They can be profitable!
To answer your questions - some of them, you can keep open, or close after 4 weeks or 30 days, it all depends on the shop.
Some companies might have an account opening shop, and if you keep it open, then you might also be eligible for things like teller shops.
They generally do not impact your score, but they may check your credit, or a system called ChexSystems. This is more of a system to look for someone who bounces checks vs. a credit score.
That said, I have opened accounts and received a bonus from the bank (you can google "bank account bonuses"
and see if there may be some that are available, no report necessary. Some of them might be more lucrative than a mystery shop bank opening.
It will depend on the company and the bank what is required of you, but hopefully that helped answer some questions. Good luck!