Probably yes and must be disclosed on your green card application. You must submit certified copies of criminal incidents and receipts of paid...

When you became an LPR you should have updated SSN, DL, etc. When you get your US passport that's proof of USC. Your 14-year-old if eligible may...

You may request premium processing of the from I485. The application will be adjudicated on its merit.

If all in your statement is accurate, this incident may not adversely impact your green card application.

Your criminal history may have an adverse impact on future immigration decision.

LPR may work anywhere, no permission is needed.

An LPR can petition for a spouse who in status on a visitor's visa.

LPR status is a prerequisite for US naturalization. Consult an immigration lawyer for further guidance.

I have moved this question from immigration to family law for appropriate answer. When resolved then immigration issue may be considered.

The time varies allow about 1 year or more.