Airlines routinely allow people to start class before they complete background checks...they don't like to spend the money on someone who may no-show their new hire class. They have to have the checks completed prior to you flying the line, so they have 6-8 weeks to get that done while you're in training (it usually takes 2-3 weeks max).

The also routinely invite people to leave during groundschool due to background problems; it's crappy since now you have to try to get your old job back...

I agree about the credit checks for the mail, that is my understanding also. Every airline carries mail occasionally. However I have experience with military security clearances and I suspect that having a lot of debt due to being a starving student/cfi is not necessarily a problem, they are looking for ridiculous debt levels relative to your station in life or massive defaults indicating drugs, gambling, or gross irresponsibility.