1. If your character is defeated and is back in town and the main quest is completed, does this character remain permanently dead?

what do you mean by "main quest"?

leoj said:

Are we talking about vanilla Descent, or one of the Advanced Campaigns? In either case, a hero is resurrected in town immediately after he is killed, as I understand the rules anyway. He then needs to wait for his next turn before he can do anything.

If it's vanilla Descent, then it doesn't really matter. The quest is over. Nothing gets carried forward to the next quest and everyone starts fresh, so whether your hero is "permanently dead" or not is irrelevant. Technically I believe you're supposed to draw all new heroes for the next quest, although a lot of people (our group included) tend to keep the same heroes and reset all their gear/training, just because it's faster or something.

If you're playing an Advanced Campaign and the party completes the final level of a given dungeon, then the hero who was resurrected in town will rejoin the party automatically when the dungeon ends.

The number of heroes in a given game is always constant. Four heroes in AC or 2-4 heroes in vanilla (4 is still recommended for the best balance.) There is no way to permanently remove one. The closest you'll get to that is if a hero is killed in an outdoor encounter in the advanced campaign, in which case he gets resurrected as normal, but without a glyph on the battle map he's unable to rejoin the party until the encounter ends.