For legal permanent residents who have had a run-in with the law, the green card renewal process will trigger a review of your criminal record by USCIS.
The I-360 application is available to select groups of immigrants as a means of gaining permanent resident status in the U.S. Here is an overview of the various eligible groups.
If you're in the U.S. on a J-1 visa, and you would like to apply for a green card or other immigration benefit, you might need to return to your home country for two years; unless you qualify for a waiver.
The terms "resident alien" and "non-resident alien" are actually from the federal tax laws. Resident aliens owe tax on all their income, while non-resident alien owe tax only on income generated from U.S. sources.