Left 4 Dead 2 Official Dedicated Bugging Going Black Screen

It is now possible to play old mutations; when you select Mutation, the last option in the menu is "Change Mutation," and there's a pretty good list of Mutations you can play, in 1-, 4-, or 8-player games.

Note that, because there are a couple of dozen available mutations (counting all modes), it can be hard to get a game going unless you're willing to wait a while or invite some friends. If you want to play soon, invite friends or get used to bots. ("Four Swordsmen" is awful with bots, by the way. If you get grabbed by a Smoker on a ledge, your teammates can't figure out how to get to his ledge and stab him.)

Yes, you can. You have to make sure you know your way around the console, and playing online might or might not work.
Basically, you have to enter these commands in the console to make it work:

ss_enable 1
ss_map "map_name"
connect_splitscreen localhost

You have to make sure your Xbox 360 controller is set up correctly, for that, read this forum post: Left 4 Dead: SplitScreen PC Guide.

It's possible play split screen with a non-Xbox 360 controller, but you'll need to set up your own controller configurations by editing edit \left4dead\cfg\360controller.cfg. The lines you'll need to edit are the ones starting with "cmd2 bind".