I worked at a Kinko's for 3.5 years, so I know all the ins and outs of the operation of such a place. Here are some pointers:
1- It's true, they will most likely not print out your covers due to copyright issues. Even if you explain any legal loopholes to them, they will refuse. They're just covering their own legal bases. This is OK though, because they will most likely get the scaling or placement wrong anyway.
2 - Your best bet, is to rent a computer and print it yourself. The color printers they have are usually of the Xerox docucolor series, which is a very good lazer printer. They will have lazer paper loaded into it, which is slightly thicker, smoother, and brighter than normal printer paper. Computer use costs about 10 cents per minute, so be quick. When I left the company, color prints were at 50 cents a page, which was down from 89 cents a page. The Fedex acquisition did a lot of things to the company, some good, some bad.
3- To rent a computer, you have to use their card system. You can either insert your credit or debit card directly into the terminal and start using the machine, or you can go to the little kiosk they'll have setup. This kiosk will let you load money onto a prepaid card. Computer use and printing will then deduct from the balance. If you run out, you are given a certain amount of time to go back to the kiosk and reload. If you have change, they'll cash the card out at the register.