Playing around with a Rubik's Cube in TikZ

For your first question, a very simple example of how the tikz-3dplot handles its coordinate changes. Note the \tplotsetmaincoords{<angle>}{<angle>} command that sets the view.

I trust you'll be able to add the colors.

\documentclass[border=5pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\begin{document}
    \foreach \myPsi in {90,100,...,170}{
    \tdplotsetmaincoords{70}{\myPsi}
    \begin{tikzpicture}
        \clip (-8,-6) rectangle (8,6);
        \begin{scope}[tdplot_main_coords]
            \draw[step=2cm,canvas is yz plane at x=4] (-4.01,-4.01) grid (4,4);
            \draw[step=2cm,canvas is xz plane at y=4] (-4.01,-4.01) grid (4,4);
            \draw[step=2cm,canvas is yx plane at z=4] (-4.01,-4.01) grid (4,4);
        \end{scope}
    \end{tikzpicture}
    }
\end{document}

Edit
This is bit more realistic with rounded corners:

\documentclass[border=5pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\begin{document}
    \pgfmathsetmacro\radius{0.1}
    \foreach \frontcolor [remember=\frontcolor as \sidecolor (initially blue)] in {red,white,orange,blue}{
        \foreach \myPsi in {90,100,...,170}{
            \tdplotsetmaincoords{70}{\myPsi}
            \begin{tikzpicture}[line join=round]
                \clip (-3,-2.5) rectangle (3,2.5);
                \begin{scope}[tdplot_main_coords]
                    \filldraw [canvas is yz plane at x=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \filldraw [canvas is xz plane at y=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \filldraw [canvas is yx plane at z=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \foreach \X in {-1.5,-0.5,0.5}{
                        \foreach \Y in {-1.5,-0.5,0.5}{
                            \draw [canvas is yz plane at x=1.5,shift={(\X,\Y)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is xz plane at y=1.5,shift={(\X,\Y)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is yx plane at z=1.5,shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                        }
                    }
                \end{scope}
            \end{tikzpicture}
        }
    }
\end{document}

Edit 2
As per request, rotating one row:

The code becomes increasingly complex, and drawing order is very important.

\documentclass[border=5pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}
\begin{document}
    \pgfmathsetmacro\radius{0.1}
    \foreach \frontcolor [remember=\frontcolor as \sidecolor (initially blue)] in {red,white,orange,blue}{
        \foreach \myPsi in {90,100,...,170}{
            \tdplotsetmaincoords{70}{100}
            \begin{tikzpicture}[line join=round]
                \clip (-3,-2.5) rectangle (3,2.5);
                \begin{scope}[tdplot_main_coords]
                    \filldraw [canvas is yz plane at x=1.5] (-1.5,-1.5) rectangle (1.5,0.5);
                    \filldraw [canvas is xz plane at y=1.5] (-1.5,-1.5) rectangle (1.5,0.5);
                    \filldraw [canvas is yx plane at z=0.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \foreach \X in {-1.5,-0.5,0.5}{
                        \foreach \Y in {-1.5,-0.5}{
                            \draw [canvas is yz plane at x=1.5,shift={(\X,\Y)},fill=blue] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is xz plane at y=1.5,shift={(\X,\Y)},fill=red] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                        }
                    }
                    \tdplotsetrotatedcoords{0}{0}{-\myPsi+90}
                    \begin{scope}[tdplot_rotated_coords]
                        \foreach \X in {-1.5,-0.5,0.5}{
                            \filldraw [canvas is yz plane at x=1.5,shift={(\X,0.5)}] (0,0) rectangle (1,1);
                            \filldraw [canvas is xz plane at y=1.5,shift={(\X,0.5)}] (0,0) rectangle (1,1);
                            \draw [canvas is yz plane at x=1.5,shift={(\X,0.5)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is xz plane at y=1.5,shift={(\X,0.5)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \foreach \Y in {-1.5,-0.5,0.5}{
                                \filldraw [canvas is yx plane at z=1.5,shift={(\X,\Y)}] (0,0) rectangle (1,1);
                                \draw [canvas is yx plane at z=1.5,shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            }
                        }
                    \end{scope}
                \end{scope}
            \end{tikzpicture}
        }
    }
\end{document}

To get it to rotate back and forth I cheated a bit when converting it to a .gif:

Edit 3
This pretty much makes you able to control the rotation with buttons:

\documentclass[]{article}
\usepackage{animate}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}

\newwrite\OutFile%
\immediate\openout\OutFile=tl.txt%
\immediate\write\OutFile{::0x0,1}
\foreach \i in {2,...,36}{
    \immediate\write\OutFile{::\i}%
}
\immediate\closeout\OutFile


\pgfmathsetmacro\radius{0.1}
\tdplotsetmaincoords{70}{100}

\newcommand{\drawRotatedRow}[2][2]{
    \pgfmathsetmacro\myHeight{-1.5+int(#1)}
    \pgfmathsetmacro\myPsi{#2}
    \pgfmathsetmacro\mySecondPsi{-80+Mod(\myPsi+80,90)}
    \pgfmathtruncatemacro\mySegment{Mod((\myPsi+80)/90,4)}
    \ifcase\mySegment% segment 0
        \def\frontcolor{red}
        \def\sidecolor{blue}
    \or% segment 1
        \def\frontcolor{blue}
        \def\sidecolor{orange}
    \or% segment 2
        \def\frontcolor{orange}
        \def\sidecolor{white}
    \or% segment 3
        \def\frontcolor{white}
        \def\sidecolor{red}
    \fi
    \begin{scope}[tdplot_main_coords]
        \tdplotsetrotatedcoords{0}{0}{\mySecondPsi}
        \begin{scope}[tdplot_rotated_coords]
            \filldraw [canvas is yx plane at z={\myHeight+1}] (-1.5,-1.5) rectangle (1.5,1.5);
            \filldraw [canvas is yz plane at x=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \filldraw [canvas is xz plane at y=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \foreach \X in {-1.5,-0.5,0.5}{
                \draw [canvas is yz plane at x=1.5,shift={(\X,\myHeight)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \draw [canvas is xz plane at y=1.5,shift={(\X,\myHeight)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \ifnum#1=2\relax
                    \foreach \Y in {-1.5,-0.5,0.5}{
                        \draw [canvas is yx plane at z={\myHeight+1},shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                    }
                \fi
            }
        \end{scope}
    \end{scope}
}
\begin{document}
    \begin{animateinline}[controls,loop,timeline=tl.txt]{10}
        \begin{tikzpicture}[line join=round]
            \clip (-3,-2.5) rectangle (3,2.5);
            \drawRotatedRow[0]{0}
            \drawRotatedRow[1]{0}
        \end{tikzpicture}
        \newframe
        \multiframe{36}{iPsi=0+10}{%
            \begin{tikzpicture}[line join=round]
                \clip (-3,-2.5) rectangle (3,2.5);
                \drawRotatedRow{\iPsi}
            \end{tikzpicture}
        }
    \end{animateinline}
\end{document}

I added a command that draws a row of cubes, with optional z level (defaults to 2, zero based) and with a rotation about z: \drawRotatedRow[<level>]{<rotation>}. With this command now we can do something like this:

\documentclass[tikz]{standalone}
\usepackage{animate}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}

\pgfmathsetmacro\radius{0.1}
\tdplotsetmaincoords{70}{100}

\newcommand{\drawRotatedRow}[2]{
    \pgfmathsetmacro\myHeight{-1.5+int(#1)}
    \pgfmathsetmacro\myPsi{#2}
    \pgfmathsetmacro\mySecondPsi{-80+Mod(\myPsi+80,90)}
    \pgfmathtruncatemacro\mySegment{Mod((\myPsi+80)/90,4)}
    \ifcase\mySegment% segment 0
        \def\frontcolor{red}
        \def\sidecolor{blue}
    \or% segment 1
        \def\frontcolor{blue}
        \def\sidecolor{orange}
    \or% segment 2
        \def\frontcolor{orange}
        \def\sidecolor{white}
    \or% segment 3
        \def\frontcolor{white}
        \def\sidecolor{red}
    \fi
    \begin{scope}[tdplot_main_coords]
        \tdplotsetrotatedcoords{0}{0}{\mySecondPsi}
        \begin{scope}[tdplot_rotated_coords]
            \filldraw [canvas is yx plane at z={\myHeight+1}] (-1.5,-1.5) rectangle (1.5,1.5);
            \filldraw [canvas is yz plane at x=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \filldraw [canvas is xz plane at y=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \foreach \X in {-1.5,-0.5,0.5}{
                \draw [canvas is yz plane at x=1.5,shift={(\X,\myHeight)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \draw [canvas is xz plane at y=1.5,shift={(\X,\myHeight)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \ifnum#1=2\relax
                    \foreach \Y in {-1.5,-0.5,0.5}{
                        \draw [canvas is yx plane at z={\myHeight+1},shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                    }
                \fi
            }
        \end{scope}
    \end{scope}
}
\begin{document}
    \foreach \iPsi in {0,10,...,359}{
        \begin{tikzpicture}[line join=round]
            \clip (-3,-2.5) rectangle (3,2.5);
            \drawRotatedRow{0}{-\iPsi}
            \drawRotatedRow{1}{0}
            \drawRotatedRow{2}{\iPsi}
        \end{tikzpicture}
    }
\end{document}

Or even this (very long GIF):

\documentclass[tikz]{standalone}
\usepackage{animate}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{3d}

\pgfmathsetmacro\radius{0.1}
\tdplotsetmaincoords{70}{100}

\newcommand{\drawRotatedRow}[2]{
    \pgfmathsetmacro\myHeight{-1.5+int(#1)}
    \pgfmathsetmacro\myPsi{#2}
    \pgfmathsetmacro\mySecondPsi{-80+Mod(\myPsi+80,90)}
    \pgfmathtruncatemacro\mySegment{Mod((\myPsi+80)/90,4)}
    \ifcase\mySegment% segment 0
        \def\frontcolor{red}
        \def\sidecolor{blue}
    \or% segment 1
        \def\frontcolor{blue}
        \def\sidecolor{orange}
    \or% segment 2
        \def\frontcolor{orange}
        \def\sidecolor{white}
    \or% segment 3
        \def\frontcolor{white}
        \def\sidecolor{red}
    \fi
    \begin{scope}[tdplot_main_coords]
        \tdplotsetrotatedcoords{0}{0}{\mySecondPsi}
        \begin{scope}[tdplot_rotated_coords]
            \filldraw [canvas is yx plane at z={\myHeight+1}] (-1.5,-1.5) rectangle (1.5,1.5);
            \filldraw [canvas is yz plane at x=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \filldraw [canvas is xz plane at y=1.5] (-1.5,\myHeight) rectangle (1.5,{\myHeight+1});
            \foreach \X in {-1.5,-0.5,0.5}{
                \draw [canvas is yz plane at x=1.5,shift={(\X,\myHeight)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \draw [canvas is xz plane at y=1.5,shift={(\X,\myHeight)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                \ifnum#1=2\relax
                    \foreach \Y in {-1.5,-0.5,0.5}{
                        \draw [canvas is yx plane at z={\myHeight+1},shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                    }
                \fi
            }
        \end{scope}
    \end{scope}
}
\begin{document}
    \foreach \level in {0,1,2}{
        \foreach \iPsi in {0,10,...,359}{
            \begin{tikzpicture}[line join=round]
                \clip (-3,-2.5) rectangle (3,2.5);
                \ifcase\level % Level 0 rotating
                    \drawRotatedRow{0}{\iPsi}
                    \drawRotatedRow{1}{0}
                    \drawRotatedRow{2}{0}
                \or % Level 1 rotating
                    \drawRotatedRow{0}{0}
                    \drawRotatedRow{1}{\iPsi}
                    \drawRotatedRow{2}{0}
                \or % Level 2 rotating
                    \drawRotatedRow{0}{0}
                    \drawRotatedRow{1}{0}
                    \drawRotatedRow{2}{\iPsi}
                \fi
            \end{tikzpicture}
        }
    }
\end{document}

Just for completeness. As there was the question about pgfplots, I just spell out Max Snippe's comment.

\documentclass[border=3.14mm,tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\def\amax{4} %number of squares in each direction
\foreach \X in {30}
{\begin{tikzpicture}
%\path[use as bounding box] (-5,5) rectangle (5,5);
\begin{axis}[height=5cm,unit vector ratio=1 1 1,view={\X}{20},colormap/hot,
    set layers=standard,
    domain=0:{\amax+1},
    domain y=0:{\amax+1},
    samples y=1,
    xmin=-1,ymax=\amax+1,
    hide axis,
    xtick=\empty, ytick=\empty, ztick=\empty,
    clip=false,samples=\amax+1,samples y=\amax+1
]
\addplot3[point meta=rand,mesh,surf] (x,y,\amax+1);
\addplot3[point meta=rand,mesh,surf] (x,0,y);
\addplot3[point meta=rand,mesh,surf] (\amax+1,x,y);
\end{axis}
\end{tikzpicture}}
\end{document}

Note:

• I was not able to do a proper animation. For some reason the plots got doubled on each slide. I have no idea what's going on, and I was able to do proper pgfplots animations in the past. Most likely I am doing something really dumb. I was really dumb. For another question I changed the viewers preference to two page view. So one could do animations but compared to Max Snippe's result the outcome will be poor.

• If you give me rough idea what the colors should be I will be happy to add them as well. Ideally this would be some cool formula such that one could use point meta for that, otherwise I guess one has to resort to tables. I added random colors from a colormap that describes the temperatures these days. If I was not hibernating in winter, I could change it to cool then.

ADDENDUM: All credits go to current_user, who had the idea, Max Snippe, who made the superb code (which I just stole), and samcarter, the author of tikzmarmots. ;-)

\documentclass[border=5pt,tikz]{standalone}
\usepackage{tikz-3dplot}
\usepackage{tikzmarmots}
\newsavebox\Marmot
\savebox\Marmot{\tikz[scale=0.4]{\marmot[whiskers,teeth]}}
\usetikzlibrary{3d}
\begin{document}
    \pgfmathsetmacro\radius{0.1}
    \foreach \frontcolor [remember=\frontcolor as \sidecolor (initially blue)] in {red,white,orange,blue}{
        \foreach \myPsi in {90,100,...,170}{ %  
            \tdplotsetmaincoords{70}{\myPsi}
            \begin{tikzpicture}[line join=round]
                \clip (-3,-2.5) rectangle (3,2.5);
                \begin{scope}[tdplot_main_coords]
                    \filldraw [canvas is yz plane at x=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \filldraw [canvas is xz plane at y=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \filldraw [canvas is yx plane at z=1.5] (-1.5,-1.5) rectangle (1.5,1.5);
                    \foreach \X in {-1.5,-0.5,0.5}{
                        \foreach \Y in {-1.5,-0.5,0.5}{
                            \draw [canvas is yz plane at x=1.5,shift={(\X,\Y)},fill=\sidecolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is xz plane at y=1.5,shift={(\X,\Y)},fill=\frontcolor] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \draw [canvas is yx plane at z=1.5,shift={(\X,\Y)},fill=green!60!black] (0.5,0) -- ({1-\radius},0) arc (-90:0:\radius) -- (1,{1-\radius}) arc (0:90:\radius) -- (\radius,1) arc (90:180:\radius) -- (0,\radius) arc (180:270:\radius) -- cycle;
                            \begin{scope}[canvas is yz plane at x=1.5,transform shape]
                            \node at (\X+0.5,\Y+0.5) {\usebox{\Marmot}};
                            \end{scope}
                            \begin{scope}[canvas is xz plane at y=1.5,transform shape]
                            \node at (\X+0.5,\Y+0.5) {\usebox{\Marmot}};
                            \end{scope}
                            \begin{scope}[canvas is yx plane at z=1.5,transform shape]
                            \node[yscale=-1] at (\X+0.5,\Y+0.5) {\usebox{\Marmot}};
                            \end{scope}
                        }
                    }
                \end{scope}
            \end{tikzpicture}
        }
    }
\end{document}

Barbara Beeton discovered that the marmots are wagging their tails. This is because Max Snippe's routines are automatically such that the tails of the marmots are always behind the marmot. There is a reflection at the right moment. Therefore one may want to promote the marmots to 3D. (I am very optimistic that the tikzmarmots package will provide 3D marmots in the near future. The impatient users may get the following animation by replacing all \fill[ commands by \shade[ball color= in that package and modify the last marmot node to \node[yscale=-1,rotate=\myPsi-90] at (\X+0.5,\Y+0.5) {\usebox{\Marmot}};.)