Sorry if I was snippy, but this isn't a drive-by homework service. If you already have proofs of the propositions you want proven, then by all means post them and highlight any parts you're not sure about. If you are having trouble getting started on the proof, it might help to tell us what sort of proof experience you have.
I...just realized that it's June. Still, it would be nice of you to post your motivation and your reasoning so far, and less likely to be mistaken for homework.
If you search for "N2-EXPTIME" including the quotes, you'll find the results I indicated, except there are four results now due to this thread.
Regarding question 3, you reversed the conditional in both the statement about P, NP, and co-NP, and in your question itself, so there was no internal inconsistency. It's not my fault that you provided absolutely no context that would let me figure out whether that was a typo or you genuinely had it backward in your mind.
OK, I realized who you are, and this isn't homework.
1. Yes, I agree.
3. Yes (the converse). A language is a set of inputs, a complexity class is thus a set of sets, and a complement class co-X is literally just a set of the complements of the elements of X. So if X = Y, co-X = co-Y is just a property of equality. But any class DTIME(f(n)) is its own complement because switching accept and reject on a deterministic TM decides the complement language without changing the running time. So ANY complexity class equal to ANY DTIME class will be its own complement by basic properties of equality.