You can shoot at anybody in an Anarchy system (not sure about in the no fire zone around the station) and not worry about getting fined. The local system won't post anybody as wanted, but you can use the warrant scanner to see if other places have bounties on them.
Yes, you can do missions for 1 faction, and get friendly with them. Even if your friendly with them, their ships might still attack you from time to time. If they attack you, feel free to fight back, I haven't noticed any loss to actually killing faction members in an Anarchy system. If you do lose any status with a faction for killing their ships in the anarchy system, it looks to be minimal.