So someone has put a dark elf Pleasure Cults quest into my battlefield. What's worse is that I'm playing 3 playa Cataclysm. So I have one of my units questing on it so I can get rid of it asap. Can this questing unit of mine defend the battlefield? Defend a legend being attacked through my battlefield? Defend the fulcrum I'm holding on to? Attack other players zones and fulcrums? I'm unfamiliar with these types of quests.

Yes to all, unless it's corrupted or anything else keeps it from doing these things. A unit on a quest, no matter the zone they're in, behaves completely normally. It contributes power in the kingdom or quest zone, can defend, can attack if it's in the battlefield or an effect allows it to attack from anywhere else, use its abilities, and so on. It's just questing in addition to all it normally does.