Remember that very soon, people will try to look you up on Google/Bing/Yahoo/etc. to get a better grasp of your research, past, etc. Facebook is not a very hard site to 'open up' and see all of those fun things about you that you don't want to be seen. Rule of thumb: if you don't want your supervisor (or other professional contacts) to see it, don't put it up online.
As for friend or not, that is a personal call. I wouldn't mind having my sup as a 'friend' as it would open up their network of other researcher friends that I might be able to do future research with, should interest align. It is a networking tool that has several elements of social life incorporated into it. But ignoring the request is probably the easiest way of avoidance. If I was you, that would be my plan, most people forget about it after a while and don't bug you with 'why haven't you accepted yet' stuff.