Is any problem with me staying out of the country while I applied for naturalization?

By returning to the U.S. before hitting 180-day mark, you technically avoided the presumption of broken residence. But the abandonment inquiry...

USCIS probably decided whatever evidence it requested in RFE was in the file (already), irrelevant, or unnecessary (at least for the time being).

You can file application (I-485/I-130) with the basic/essential documents (i.e., without bona-fides). Focus on submitting the bona-fide evidence...

It's legally (theoretically) possible, but it will be an uphill case (and it's your burden to overcome the presumption that your continuous...

As long as you could show that you already took care of those traffic tickets, your N-400 application should be fine.

How do I extend my green card re-entry permit?

You could not "extend" your re-entry permit. But you could re-apply for a new re-entry permit, assuming you still have un-relinquished residence...

Consult with an attorney to see if your removal proceedings can be terminated on time (so you won't get an in-absentia removal order for not...

No, those benefits for children will not affect your green card (or the pending I-751 case).

Police reports (or calling police on the abuser) are not required for VAWA eligibility. You could provide your own affidavit/declaration detailing...

What do I need to give the US. Passport agecy?

You must be trying to show automatic acquisition of U.S. citizenship (probably under CCA of 2000). It could be anything that lists you (or tends...