I think if you call Amex you may be able to do a postpaid MGM booking. Note that typically all postpaid MGM stays (booked through any channel) have a deposit of one night taken at booking and the remainder charged at check-out, though it appears the Amex offer allows for the spend to be accumulated across multiple transactions so that should be okay. (A prepaid booking will show up as an Amex Travel charge, not an MGM charge, so wouldn't count toward the Amex offer at all, though you could potentially still achieve the Amex offer on a postpaid booking if you had a lot of incidental spend charged to your room -- though obviously that may be hard on a short stay after taking advantage of the FHR inclusions, but you can at least charge to your room from restaurants at any MGM property on the Strip.)

There is a whole other thread on the second question; short answer is YMMV but changing rooms is the easiest way to make sure that you get both credits. FHR back-to-back booking questions.