How long does it normally take before the shop you submit gets either approved or denied and how long after that does it take to receive payment for the job?

It depends on the MSP they each have this info on the contracts though.

It varies widely. I did some a few days ago and have already been paid for them. But payment could take weeks.

Expect some to take up to 4 weeks, and some a few days. It depends mainly on the payment schedule of the company you are doing it for. Some companies pay out every two weeks, and some only once a month. Look at your contract for the job or on the companies website FAQ pages.