If the school is not fully accredited and has no ranking, is it still possible to transfer out to a 'higher ranked' school? Will any schools, let alone top schools, entertain the possibility of a transfer from UCI or is that not possible?

UC Irvine is an odd beast. I think it warrants a phone call to the law schools you wish to apply.

I remember reading that students at provisionally accredited schools have all the same rights as students at full accredited schools, but I'd say that warrants a phone call to be sure.