I'm a big PvP nut. So besides in Cyrodill, where will we be able to PvP? Will there be PvP in non-instanced dungeons or in "battlegrounds" where there objectives (capture the flag, deathmatch, etc). Basically, what will be all the ways we can PvP besides in Cyrodill?

Only in Cyrodiil.

If 1v1 fights are in, then anywhere. Otherwise, Cyrodiil will be the only place to fight other people.