The form you are referring to is the *federal* 4473 form. The "yes" questions are the first and last.
The backgroud check is the federal NICS system. Your name and license number will be entered (either manually or via swiping your DL) into the computer, and they will look for any past felonies, domestic violence, court-ruled mental instabilities, or things like "failure to appear." Misdemeanors other than domestic violence won't prevent you from buying a gun.