How to get information about remaining free space?

I think you could use \pagetotal for this. Subtract it from \textheight to get the amount of space remaining.

\documentclass[10pt]{article}
\newdimen\spaceleft
\spaceleft=\textheight
\multiply\spaceleft by -1
\usepackage{lipsum}
\begin{document}
\lipsum[1-3]
\advance\spaceleft by \pagetotal
\multiply\spaceleft by -1
\typeout{Space on last page:}
\showthe\spaceleft
\end{document}

The result of this is:

Space on last page:

204.0pt.

A nicer way to realise Ian Thompsons answer is to simply use \dimexpr rather than multiplying by -1 and adding the lengths:

\documentclass{article}
\usepackage{lipsum}

\begin{document}
\lipsum[1-3]

\newdimen\spaceleft
\spaceleft=\dimexpr\textheight-\pagetotal\relax

Remaining space (above this line):
\the\spaceleft
\end{document}