Showing planes in 3D

I apologize in advance for deleting most of the code that you show, but I really cannot understand how it relates to the question. For the time being this answer only defines a way to create blurry planes.

\documentclass[t]{beamer}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{shadows.blur}
\definecolor{mygreen}{RGB}{155,183,174}
\makeatletter
\tikzset{render blur shadow/.code={\pgfbs@savebb%
\pgfsyssoftpath@getcurrentpath{\pgfbs@input@path}%
\pgfbs@compute@shadow@bbox%
\pgfbs@process@rounding{\pgfbs@input@path}{\pgfbs@fadepath}%
\pgfbs@apply@canvas@transform%
\colorlet{pstb@shadow@color}{white!\pgfbs@opacity!#1}%
\pgfdeclarefading{shadowfading}{\pgfbs@paint@fading}%
\pgfsetfillcolor{#1}%
\pgfsetfading{shadowfading}{\pgftransformshift{\pgfpoint{\pgfbs@midx}{\pgfbs@midy}}}%
\pgfbs@usebbox{fill}%
\pgfbs@restorebb%\globalcolorstrue
},render blur shadow/.default=shadowcolor,
shadow color/.code=\colorlet{shadowcolor}{#1},
blurry plane/.style={transform shape,fill=#1,shadow color=#1,rounded corners=2pt,
blur shadow={shadow blur steps=25,shadow scale=1.1,shadow opacity=100,
shadow xshift=0ex,shadow yshift=0ex,shadow blur radius=1.5ex}}}
\colorlet{shadowcolor}{black}
\makeatother
\begin{document}

\begin{frame}
\frametitle{Blurry planes}
\begin{center}
\tdplotsetmaincoords{70}{110}
\begin{tikzpicture}[tdplot_main_coords,line cap=round,>=stealth]
   \begin{scope}[canvas is zx plane at y=0]
    \path node[anchor=south west,minimum size=5cm,blurry plane=mygreen]{};
   \end{scope}
   \begin{scope}[canvas is zx plane at y=3]
    \path node[anchor=south west,minimum size=5cm,blurry plane=mygreen]{};
   \end{scope}
 \end{tikzpicture}
 \end{center}
\end{frame}
\end{document}

Here is a version that draws the planes through some points. It requires the 3d tools library.

\documentclass[t]{beamer}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{shadows.blur}
\definecolor{mygreen}{RGB}{155,183,174}
\makeatletter
\tikzset{render blur shadow/.code={\pgfbs@savebb%
\pgfsyssoftpath@getcurrentpath{\pgfbs@input@path}%
\pgfbs@compute@shadow@bbox%
\pgfbs@process@rounding{\pgfbs@input@path}{\pgfbs@fadepath}%
\pgfbs@apply@canvas@transform%
\colorlet{pstb@shadow@color}{white!\pgfbs@opacity!#1}%
\pgfdeclarefading{shadowfading}{\pgfbs@paint@fading}%
\pgfsetfillcolor{#1}%
\pgfsetfading{shadowfading}{\pgftransformshift{\pgfpoint{\pgfbs@midx}{\pgfbs@midy}}}%
\pgfbs@usebbox{fill}%
\pgfbs@restorebb%\globalcolorstrue
},render blur shadow/.default=shadowcolor,
shadow color/.code=\colorlet{shadowcolor}{#1},
blurry plane/.style={transform shape,fill=#1,shadow color=#1,rounded corners=2pt,
blur shadow={shadow blur steps=25,shadow scale=1.1,shadow opacity=100,
shadow xshift=0ex,shadow yshift=0ex,shadow blur radius=1.5ex}}}
\colorlet{shadowcolor}{black}
\makeatother
\usetikzlibrary{3dtools}
\tikzset{orthonormal basis vectors of plane/.code n args={3}{%
\pgfmathsetmacro{\mya}{TD("(#2)-(#1)")}
\pgfmathsetmacro{\mynormalization}{1/sqrt(TD("(\mya)o(\mya)")}
\pgfmathsetmacro{\mya}{TD("\mynormalization*(\mya)")}
\pgfmathsetmacro{\myb}{TD("(#3)-(#1)")}
\pgfmathsetmacro{\mynormalization}{1/sqrt(TD("(\myb)o(\myb)")}
\pgfmathsetmacro{\myb}{TD("\mynormalization*(\myb)")}
\pgfmathsetmacro{\myn}{TD("(\mya)x(\myb)")}
\pgfmathsetmacro{\mynormalization}{1/sqrt(TD("(\myn)o(\myn)")}
\pgfmathsetmacro{\myn}{TD("\mynormalization*(\myn)")}
\pgfmathsetmacro{\myc}{TD("(\myn)x(\mya)")}
\tikzset{insert path={
 (\mya) coordinate(\pgfkeysvalueof{/tikz/3d/projections/planex})
 (\myc) coordinate(\pgfkeysvalueof{/tikz/3d/projections/planey})
 }}},
3d/projections/.cd,planex/.initial=planex,planey/.initial=planey}

\begin{document}
\begin{frame}[t]
\frametitle{Blurry planes}
\tdplotsetmaincoords{70}{160}
\begin{tikzpicture}[tdplot_main_coords]
 \path (0,0,0) coordinate (O)
    (2,-2,2) coordinate (v) 
    (0,2,2) coordinate (w) 
    (-3,5,2) coordinate(m)
    [3d coordinate={(v')=(v)+(m)},
    3d coordinate={(w')=(w)+(m)}];
 \path[orthonormal basis vectors of plane={O}{v}{w}]; 
 \begin{scope}[x={(planex)},y={(planey)},canvas is xy plane at z=0,
    transform shape]
  \path node[anchor=south west,minimum width=3cm,minimum height=9cm,
    blurry plane=mygreen]{}
    (m) node[anchor=south west,minimum width=3cm,minimum height=9cm,
    blurry plane=mygreen]{};
 \end{scope}
\end{tikzpicture}
\end{frame}
\end{document}