The background is completed while you are in indoc. Finger prints are done the first day along with badges. Disclose everything on the airlineapps application or at the very latest at the interview. The airlineapps they have is a snapshot of what your application looked like when you recieve the personality test. Anything other than minor corrections at the interview is a bad thing. They also have had people not resign from other airlines while in indoc and it didn't end well.
For anyone starting the process at United you need to look over your airlineapps with a fine tooth comb BEFORE they select you. Update it regularly and accurately. Make sure no weird boxes are checked that don't apply to you.