Yes. You would be able file my green card in EB 1C category from employer.

An honest error is never grounds for adverse decision or denial. There would be an opportunity to make corrections.

Yes, a birth certificate is required to start most USCIS applications.

I think it is best to consult a lawyer in private and explore options you may be eligible to pursue.

The last action is the approval of I-130 and transfer to NVC.

I would request application to be expedited with medical supporting documents.

Yes, you may. I am unaware of any rule that invalidates an AP because photo is smudged. The photo on the DHS system, which you don't see is not...

Yes, an LPR can have a travel document with re-entry permit.

You may file the I-130 and I-485 concurrently for AOS.

You cannot file for K-1 until you divorce is final.