Yes, the non-home country makes the decision about H-1B stamping. Contact specific embassy or consulate general for requirements.

At 34 you too old to register. If you were born in the US, attended high School and/or college in the US, you are probably registered, without...

It depends on the nature and circumstances of the trip outside the US. Consult with a lawyer, share and discuss details.

It's up to the third country embassy/consulate general to accept or not accept visa applications.

Without more, a USC may petition to sponsor a FN spouse.

Additional information is required. Include your clear immigration status in the US and purpose of mom visit to the US.

Yes, you may renew work permit (AOS THROUGH VAWA) while I-601 is pending. Good luck.

This is not an immigration issue. Contact your local vehicle registration office for requirements.

Additional information is required. What is the underlying basis for filing I-485??

In general, yes. You may invite mom on green card for wedding, if mom has a valid green card. Otherwise, a visa is required.