No they will need the I601A waiver to be approved.

Yes you may travel with AP to return to the US.

Your inquiry is complex and requires a consultation with an attorney for best guidance.

You would still to comply with country visa requirements.

Yes this is possible but a cosignor may be better.

Yes this should be possible with valid visa.

No there is no option available for family based processing.

Yes the N400 can be filed with the I751 pending.

Either address should suffice for this purpose.

The passport should suffice but you can review your GC filing for certificate.