How can I fix papers?

No, not unless you are eligible under section 245(i). Consult with an experienced immigration attorney to see what options are available to you.

How can I fix my papers?

As the spouse of a USC your husband may file an immediate relative petition on your behalf. Once approved, assuming your only admissibility issue...

I-130 Approved. What should I do next? Can I start some paperwork while I wait?

Yes, you should be preparing for his waiver as that is likely your next step, unless he is 245i eligible.

Am petitioning for my husband, do I need to have cash or assets to support my AOS

No need to list assets since you have a co-sponsor.

Yes, you may stay in the US during this time.

If you are referring to processing times, then the answer is no.

F2A category

No, you are not able to expedite the case. You will need to be patient and wait for the priority date to become current.

Do I need an Immigration attorney?

My colleague is absolutely correct.

Will there be any problems of "overstaying" (6 months) if I get married to a US citizen in May if my visa expires June 28th

Based solely on the facts presented, the overstay should not be an issue.

Im US citizen and my husband is on H1B i want to apply greencard for him but i dont work right now so who else can sponsor?

Yes, you may use his income on the affidavit of support.