Is it possible the following scenario: I want to access a third-party site through my web server. Is it possible in the middle to rewrite all request to the third party .CSS files to files on my server?
Yes, though it will be YOUR web-server doing the fetching not the client. Doing so in a badly configured way opens you up to being an open-proxy on the net which is a Bad Thing. Unless you have SSL considerations, you'll probably get better client-side parallelization by calling the css files directly on the remote server from your own pages.
This will leverage mod_rewrite and mod_proxy to do exactly what you want, requests for pretty.css on your site will be fetched from the remote site and served as if it were yours.
Where you can get into trouble is when you start introducing regexes into the RewriteRule.
This will pass anything like http://example.com/css/../cgi-bin/mt.cfg to the remote site, and the remote site will serve it up just fine.