In latex.ltx, the \footnoterule is called at the end of the minipage environment when there are footnotes. So first I redefine the \endminipage command to omit the \footnoterule.
In article.cls, the indentation is produced by the \@makefntext command. This same command is called for both main-text and minipage footnotes. So we have to redefine the \mpfootnotetext command so that when it is called, it then redefines \@makefntext.
This might be dangerous, but it seems to work. I welcome feedback from those more knowledgable.
\documentclass{article}
\usepackage{lipsum}
\newlength{\mpparindent}
\setlength{\mpparindent}{0.5in}
\let\oldendminipage\endminipage
\def\endminipage{\let\footnoterule\relax\oldendminipage}
\makeatletter
\let\@oldmpfootnotetext\@mpfootnotetext
\def\@mpfootnotetext#1{%
\renewcommand\@makefntext[1]{%
\hspace*{\mpparindent}
\parindent\mpparindent
\noindent
\hb@xt@1.8em{\hss\@makefnmark}#1}
\@oldmpfootnotetext{#1}}
\makeatother
\begin{document}
\lipsum[1]
Table~\ref{table} shows a palindrome.\footnote{You can use it to make a magic square.}
\begin{table}
\caption{Sator arepo}
\label{table}
\begin{minipage}{\textwidth}
Sator arepo tenet opera rotas.\footnote{This is a palindrome.}
\end{minipage}
\end{table}
\lipsum[2]
\end{document}
