I'd have to double check, but yes, pretty sure it's 5 weeks and not 6 weeks. Yes, it's full time during training, either Mon thru Fri or Tue thru Sat from 9am to 6pm.
I know how you feel about being nervous and the long wait. Sucks, but hang in there. If you made it to the point where they queue up a background check order, it isn't likely that they would pay for that if they weren't highly interested in you.
The background check process can be nuts. I strongly recommend following up with HireRight directly. Your file will be passed along thru many hands in their system. Call often for status updates, and ask them if there is anything you can do to help them. Always be nice to them regardless of how frustrated you may feel. Be specific with your follow up questions, such as is there any element within this process that is causing a delay? Or what element so far in this process is pending? Last part is to be ready to submit any documentation via email they ask for as quickly as possible. Also, be honest with their questions but don't give them any unnecessary information that can complicate or make the process more difficult than what it already is.
Congrats so far... I think you can relax a little bit. Where you are at in this stage is very good. But I know how you feel, it's not a done deal until you have an official offer. But still, you're doing just fine. You might be bumped to 12/3, but I am not qualified to say that. That assumption is based on reading what others have said during their timeline of events / statuses, etc.