You will need to have a technician scan the computer for codes. It is very possible that you have an oxygen sensor problem.
If you were charged criminally it will always remain on your record (I am uncertain as to whether a criminal DUI offense can be expunged or not - you would have to check on this) - if it was charged as a traffic offense it WILL always remain on your driving record. Your driving record is a COMPLETE history of your driving life and does NOT go away.
There could be an issue with a sensor which could be making your check engine light remain lit. You would be advised to see if your local mechanic can help fix the problem.
why does the engine light remain on after starting
If you can hear a click from the starter, but it does not spin, you may have sucked water into the engine. That can destroy the engine. You need to remove the spark plugs to check for water in the cylinders.
yes
engine stalls sometimes driving
Check your sensors.
The Check Engine light comes on when the ECU detects a problem with the emissions system. Seek professional help.
If the check engine light is flashing as you are driving it means you are about to destroy the catalytic converter(s). Stop the engine and get it to a garage.
Check engine oil when you have just finished driving. Check Transmission fluid when the car is running and the transmission in in N (manual) or P (automatic).
have vehicle scanned to determine the problem
Depends on the vehicle, but most of the time no.