Without knowing DL's policies, I would expect them to verify some of the information on your application/resume and google you (either before or after the interview, depending on the number of applicants) before extending a formal offer. The background check could happen after you have accepted a conditional job offer and delay your start. You probably need to sign your permission for the background check, but this might be included on the application form. You might not be able to start, or even begin training, before the results of the mandatory drug test.

Most employers will respect you for delaying the start date to give fair notification (typically one pay period by convention, but this expectation is stated in some employee handbooks) to your current employer.