• Does RedPen check LaTeX documents for common style errors?
No. RedPen itself does not support LaTex format, but you can apply RedPen to proofread LaTex documents converting them to Markdonw format. Pandoc supports the conversion from LaTex to Markdown format. The following is a sample to run RedPen to a LaTex document.
$pandoc content.tex -s -t  markdown_mmd -o content.md
$redpen -c conf/redpen-conf.xml content.md