Denied naturalization after adoption.after the child has graduated from high school and college in this country.

We would need to see the denial to see if the Uscis was in error. Without their reasoning we cannot even guess what they are thinking

Is it possible to put the I-751 petition on hold after the Filing?

No she cannot hold it but she could have written or spoken to Uscis

Yes you should be ok with the current one and an explanation

Yes! Your card was approved! It should arrive shortly.

Can a person be banned from entering the US?

It all depends on whether or not he is found to have entered into a fraudulent marriage with the first wife!

I have an approved I-130 and..................

No you would not start over. But you would need to have the USCIS transfer the file if you change your address.

What forms and documents do I need to submit at USCIS office for my husband who is legally visiting the US?

You need to file an entire package with your petition and his adjustment of status applications, medical and photos.

What paperwork do I need to prove that I completed the 2 year residency requirement associated with my J visa?

Everything you mentioned plus employment records and whatever else you can document from your time abroad.

Immigration and using insurance to cover for a birth when on a visa

Accidentally? Was it insurance or public benefits?

H1b status, visa, and PERM processing

You should extend for an additional three years. You can only travel if you have a valid visa or will apply for an extension abroad.