Depends on the state. Some states do do a criminal background check on anyone working in a hospital, especially those who might have contact with children.
For New Jersey, I not only had to be fingerprinted but had to have a criminal check, particularly looking for sexual offenses and offenses against children. There was no drug test or other screens outside of the routine vaccination titres, but I would not assume that the background check is simple. Some states and institutions (ie, especially if you are going to be rotating at a VA hospital) do require drug screens.
The key is do not lie...if you have committed a felony and the application for your license asks, tell the truth. If they find out and you have lied on your application, you can be turned down for a license and have your offer for a residency position revoked. Same for misdemeanors - generally license applications only care about felonies or non-traffic related offenses. Drug and alcohol offenses count and can also be a reason to refuse to issue a license (although most states are keen to work with physicians who have this on their record, so its not an automatic).
As for the rest of the stuff, don't start worrying about LORs and PSs until 3rd year.
