Adsense is part of your overall gmail account, banning of adsense should not affect the rest of gmail applications.
I have a proxy and ive been told that some people have been banned from having ads on their proxy page. Im just a little worried because I rely on Gmail alot.
Lol, I wouldn't worry about gmail account being banned, coz you can make endless gmail accounts. But you cannot apply for an adsense account once you're banned
If you're worried about being banned because of your proxy site, take the ads off it. Make a different site for Adsense.