You can remain outside until the travel document validity.

Yes you can return a week before expiration of the travel document.

No that shouldn’t detrimentally impact your case.

You can amend the record at time of interview.

Once your mother becomes an LPR, she can petition for the children.

May be better to obtain an I551 stamp.

You should continue to contact USCIS although it can take a few months for card to be produced.

It depends on the time it took for the I130 to be approved. Consult with an attorney for guidance.

If through spouse, you may apply after 3 years from obtaining your residency.

You must answer as truthfully as possible and provide corresponding documents to explain the incident.