Different airlines can do it differently.
The ones that are cost conscious (esp. regionals) will do it after class starts so they don't spend the money on someone who might not show up. They don't mind if they have to fire you, and are not bothered by the fact that you already resigned your previous job.
NDR should require your written authorization. A credit check might not require your authorization but I suspect most employers would obtain it anyway just to avoid any confusion.
A background check generally does not require your authorization. In some states it may, and they may need your authorization if they use a background check service (as opposed to doing it in-house).
If you're playing games with your career, you need to find out the specific practices of the company in question, and the specific rules of the state where the interview occurs.