Should we use "Married name" or "Maiden name" for the AOS (Adjustment of Status)?

Use her current legal name which appears to now be her married name. Use her maiden name where it asks for maiden name and as 'other names used.'

Does does a fiance visa need a sponsor?

If the income is not sufficient a sponsor will be required. You should consult directly with an immigration attorney for best results.

This is not an immigration question. I will re-tag under education law.

Your situation could but does not have to be 'fatal' for the underlying asylum claim. Not trying to scare you but you will really need to prove...

If you had chosen legal representation your immigration lawyer would have a full copy of what was submitted. At this point unless you yourself made...

Can I apply for re entry permit I-131 bafore traveling to the US and come to the biometric appointment?

https://www.uscis.gov/sites/default/files/USCIS/Resources/B5en.pdf At a minimum you'd need a receipt notice, and you risk not having the case...

Petition for spouse

Sure, if you are a citizen. What was your status when the petition was denied? The unauthorized work will definitely be on her record so you'd have...

1-601 and 1-212 waivers after ten years?

You overstayed your visa and then married a U.S. citizen? It sounds like your case was mishandled from the start if your most recent entry was...

A shoplifting charge as a theft crime can be interpreted that you do not possess a good moral character and may require a waiver down the line. In...

Filing I-130 I-485 I-765 I-864

Your question shows why most people in this situation should hire an immigration lawyer. Consider it.