High school transcrips are not a problem since there may be more people with your name. And they do not search in school databases for criminals. You can also have an ITIN number for taxes purposes, and they dont search the databases.

Only specialized agencies such as the FBI, DEA, CIA or the like can have a judge's permission to search for private information of that kind. Immigration might do so, if there is an open case against you, and a prosecutor is trying to find proof of your identity.

Immigration would search for crimes somebody under your name might have done in state and federal felony databases. If you were put in jail for some time and had your 10 fingersprints recorded, then you'll have a problem. For your case in which only one fingerprint was recorded, I'm not sure. I got tickets while I was in the US, but never they got my fingerprint. Ask an attorney for that specific question. Maybe you'll not be penalized with a 10-year ban.

A friend of mine was undocumented, she declared she had been illegal in the country when she was interviewed by immigration in Juarez, Mexico. But she left the country voluntarily, was not deported. She was trying to adjust her status since she married an American citizen. She said she was sorry for entering the country illegally and that she had never been in jail, and was never deported. She got her green card.
So, there are many ways to go around immigration.