Worries over background check

I just got an offer for an engineering internship and my only previous work experience I listed was for a math tutoring job I had, I listed my start date a semester earlier than it was on accident, in addition I have 2 misdemeanors for a single DUI incident where no one was hurt around 2 years ago, I have since then gotten my license back and completed all court mandated procedures and completely abstained from booze. How likely is it that I get flagged in the background check (done by a third party company) for the employment discrepancy and how likely is it my offer will get rescinded for these incidents.

I listed my start date a semester earlier than it was on accident

Contact your potential employer and correct that ASAP.

I have 2 misdemeanors for a single DUI incident

The big question here is whether you lied or not. If there was a box on a form where it said "do you have any convictions?" and you said "no", then I'd expect your internship to be dead in the water; nobody wants a liar.

If nobody asked, you haven't done anything wrong in terms of your application.

how likely is it my offer will get rescinded for these incidents

Nobody here can answer that. Contact your employer about your start date and wait and see.