adaptively sized/positioned arrow symbol pointing to the next line

You can use \hookleftarrow or carriagereturn (dingbat} as in

\newcommand\nextline{\raisebox{-1.25ex}[0pt][0pt]{$\hookleftarrow$}}
\newcommand\nextlines{\raisebox{-1ex}[0pt][0pt]{\small\carriagereturn}}

A MWE:

\documentclass{memoir}
\usepackage{dingbat}
%\usepackage{scalefnt}
\newcommand\nextline{\raisebox{-1.25ex}[0pt][0pt]{$\hookleftarrow$}}
\newcommand\nextlines{\raisebox{-1ex}[0pt][0pt]{\small\carriagereturn}}

\begin{document}

This is text at normal size. \nextline \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par

\begin{small}
This is text at small size. \nextline \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
\end{small}

\begin{footnotesize}
This is text at footnote size. \nextline \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
\end{footnotesize}

This is text at normal size. \nextlines \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par

\begin{small}
This is text at small size. \nextlines \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
\end{small}

\begin{footnotesize}
This is text at footnote size. \nextlines \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
\end{footnotesize}

\end{document}

The reason behind the strange behaviour is that you are not ending the paragraphs inside the groups that are affected by the font size switches, so the \baselineskip applied is the same as in \normalsize and this produces the inconsistency mentioned; as soon as you end the paragraphs inside the groups, the proper value for \baselineskip is applied and the problem disappears:

\documentclass{memoir}
\usepackage{amsmath}
\usepackage{scalefnt}

% in order to not load package "mnsymbol":
\DeclareFontFamily{U}{MnSymbolA}{}
\DeclareFontShape{U}{MnSymbolA}{m}{n}{
  <-6> MnSymbolA5
  <6-7> MnSymbolA6
  <7-8> MnSymbolA7
  <8-9> MnSymbolA8
  <9-10> MnSymbolA9
  <10-12> MnSymbolA10
  <12-> MnSymbolA12}{}
\newcommand*{\lcurvearrowdown}{\mathrel{\text{\usefont{U}{MnSymbolA}{m}{n}\symbol{187}}}}
\newcommand*{\nextlineref}{\raisebox{-1.25ex}[0pt][0pt]{\scalefont{1.25}\(\lcurvearrowdown\)}}

\begin{document}

This is text at normal size. \nextlineref \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx

{\small
This is text at \verb+\small+ size. \nextlineref \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
}

{\footnotesize
This is text at \verb+\footnotesize+ size. \nextlineref \\
This is more text. xxxxxxxxxxxxxxxxxxxxxxx\par
}

\end{document}

By the way, the font size switches are commands with no arguments, to be used as, for example, {\small some text\par} (the braces are to keep the change local and \par just in case more than one line is used and the proper value for \baselineskip is applied).