Yes, try quake live.
http://www.quakelive.com/
Chat with our AI personalities
I find a reasonable game (It is hard to find a great online FPS without downloading) to be a game called Russia's Army. It has 8 player online with a selection of weapons and a nickname. You can find it here: http://www.gamevial.com/russiasarmy.htm
quakelive.com is one of the easiest fps online PC games that you can play.
Halo 2
Not necessarily. It depends on how many you are trying to download. I assume by 'fps' you mean first person shooter, and in this case if you were to download quite a lot, yes. It will clutter up your hard drive and cause countless registry errors. However, if you have a fairly substantial hard drive (at least 180GB) then you are unlikely to encounter many issues. A free registry cleaner such as Eusing Free Registry Cleaner can clean up many missing or invalid entries, but then again you should check how much memory and graphics capacity is required to play your game.
One game on Facebook that is a FPS game is Paradise Paintball or now known as UberStrike Another one I know of is Brave Arms I hope this helps and red crucible