You should check the Reciprocity Tables for what you are required to produce, at travel.state.gov. If not every birth is registered in Syria, then...

Most types of fraud can be waived. There is no waiver for false claim to U.S. citizenship. If you need a fraud waiver, then file for one.

There is no way to be sure that you are safe. If you have supported designated terrorist groups in the past, you should stop immediately.

Yes, you can apply for humanitarian parole if they refuse to issue him a tourist visa or if he is ineligible for one, or if it would not come fast...

Yes. Follow up with USCIS after 30 days. Nothing you can do, be prepared that it may be misdelivered.

Schedule A does in fact require the employer to file an I-140 for you, so yes, that is your best option.

Stay in the U.S., do not travel outside to any country at all. We do not know who will be included in the ban, nor which countries, so just do not...

No, an asylee can file for adjustment of status even if he or she entered without inspection and even if he or she accrued unlawful presence.

The State Department does often consider it fraudulent to change status in the U.S. within 90 days of arriving. If she came as a tourist but was...

Yes, you qualify for naturalization even if you were removed once and you applied for, and received, a waiver and immigrated after that.