How far in advance does one have to apply for citizenship before their greencard expires?

Citizenship takes 4 months to a year.

You should file as "married" See an accounting about any other questions.

Perhaps the approval was a mistake.

H-1B transfer from university to company: a. Anytime - yes/no? b. Subject to cap - yes/no?

It depends on the employer, whether it is subject to the cap, generally it is.

My boyfriend who is currently the only one working, is on parole in 7 months he got 2 d.u.is but the second one he flipped and t

This is an immigration section, you should repost under criminal defense.

If he has proof that he is now an H1b, his wife can file for an H4.

What i have to do?

It is not possible to tell from your inquiry what information you need, so perhaps you should see an attorney.

My wife is USA Citizen and my status was H1b, she file petition for my greencard which is approved.

If you are already a permanent resident, they will not send you a work permit, advance parole, since those applications are now moot.

Naturalization application - Background check - How long it takes?

3 months is about average. Some cases never clear and require litigation.

H4 to H1 transfer

There are no visas available, you can file on April 1, 2014. There is no reason there should be a lower approval rate for H4 to H1, perhaps more...