• Does RedPen check LaTeX documents for common style errors?

No. RedPen itself does not support LaTex format, but you can get RedPen to proofread LaTex documents by first converting them to Markdown format. Pandoc supports the conversion from LaTex to Markdown format. The following will convert a LaTex document

to Markdown format, and then run redpen over the converted document.
$ pandoc content.tex -s -t  markdown_mmd -o content.md
$ redpen -c conf/redpen-conf.xml content.md