How to apply for my wife's old i797 document?

Talk to your company's immigration attorneys please.

Should we send out the form to renew his resident card

Talk to an experienced immigration attorney asap. They should be able to guide you regarding the implications of the multiple DUI's.

What's the difference between a Visa, Green Card, Citizenship, Etc. My second question, is how can they get started

There are different types non immigrant visas that allow individuals to work or study for definite periods. The green card provides lawful...

What should I do if I suspect I was out of status for 2 months?How does leaving and re-entering helps?

Based on your information you should be ok as your out of status time seems to be less than 180 days.

It depends on whose wedding it is. Talk to an immigration attorney.

What do I need to do if I didn't receive my green card and the stamp on my passport expires in 3 weeks

You can get another stamp when you make an info pass appointment. Ask about the status of your card at that time also.

What legal action can I take?

This is a family law question. Please talk to a family law attorney in the state you reside.

How to respond to the RFE, the situation is little complicated I need help .

I would talk to an attorney to assess your eligibility to adjust and also to help with the RFE if viable.

What would you suggest as to prove my single status?

You may be able to secure a single status affidavit or certificate of non-impediment from the State/City you reside in the US (some cities and...

Does my employer submit a new LCA or alter the previous one when renewing my visa?

Please discuss this with your company's immigration attorney.