If she is eligible for adjustment of status, file the I-485 application after she has been in the US at least 90 days. Then the process will occur...

Assuming no other grounds of ineligibility, a “visa overstay” should be forgiven for an immediate relative.

If you move and you file the AR-11 change of address form, your case will move to your new jurisdiction.

If you pursue an adjustment of status case based upon marriage, you would need to file a Form I-130.

If you’re planning to remain abroad in excess of one year on a particular trip, you should apply for the re-entry permit.

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It is better to submit everything to USCIS in one package.

Consider seeking a B1/B2 visitor visa instead. The issue is getting an appointment at the Consulate.

This shouldn’t be an issue. It would be good to have other evidence of the correct date.

Assuming there were no prior status violations as a F-1 student, you should be able to reenter the US on a different visa in the future.

Withdrawal will not affect a future application, but there will not be a refund of the filing fees.