Is there any software that allow us to choose which website to proxy and which one not commonly for Firefox/Chrome/IE/Other software? Is this possible with configuring the IE connection options?
Try FoxyProxy on FireFox (I think it works with other XULRunner-based apps, too). You can get as sophisticated as you'd like with it, or stick to very simple/simplistic rules. I've been using it for years with very, very few issues.