1. Complete list is: Capture, Discard (ability), Remove From Game, Return (to hand), Shuffle, Set-Aside, Take (to hand), Topdeck, and Underdeck.
2. Kram is correct on the current ruling, though this one I could see debate on. If a character defeats you by the numbers, it is technically game rule that discards you, though the character is the one defeating you. This is my understanding of the current rule, and I don't think anyone would currently argue that you could instead the discard by numbers, though I can see the logic in that as well.
3. Kram's corrected answer is correct, they can underdeck themselves as they cannot be protected against themselves.
4. Kram is correct.