Undocumented for 15 years, married to a US citizen for 10 years. Is it possible to adjust my status? Thank you for your advice.

Only if you entered with inspection and not on C/D-1, K-1, or J-1 subject to 2-year rule visas. Also, you didn't say anything about your criminal...

``IMMIGRATION FEES PAYMENT WITH PERSON BANK CHECKS`

paying with a personal check is totally OK. For #21 put to: present

There is no premium processing for I-485. EB-2 petitions are eligible for premium processing unless NIW.

Can I apply for a student visa to the US while I am in Buenos Aires (through my Syrian passport)?

For the US Consulate in Buenos Aires to adjudicate your application you need to show that you are more than just a tourist in Argentina. Your ID...

Can I apply for residential status?

Without immediate relatives no. Besides, you have omitted important information such as manner of entry, criminal record, etc.

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Immigration

She also needs a valid passport to travel. If she had been out of status by the time she filed I-485 she should abstain from traveling as some...

Whimsy (Indiscipline culture) of Embassy of Other Country located in the United States …

We all get what we deserve in terms of the government. Other than that, US has no power to intervene in this matter. Luckily!

I94 stamping and extension for a valid l1a visa. Do I need to travel to my home country?

You still need a valid passport to travel. So, why not have the visa stamped in your new passport?

Does Attempting To Enter The USA on A Fake Visa render Inadmissible?

It is unlikely she will be issued a new visa in light of the previous fraud. Also, from what you described, she is not eligible for any waiver.

Can I get my U.S. Citizenship with an expired Green Card?

You don't need the actual piece of plastic to get naturalized but it would be wise to have one in case you don't for whatever reason.