I assume you're driving since you're planning on bringing a fridge. If so, even if ASMo didn't want you to bring your own fridge (which I can't imagine why they'd care), you can drive tot he parking lot nearest to your room and then just bring it in. You don't have to go past the front desk with your bags, unless you choose to bring them with you to check-in, but I can't imagine why you'd want to carry the fridge all around the resort, so I'm guessing you'd leave it in the car until you are ready to bring your bags to your room.
For a room, really I think any room in Toy Story is a good choice. We stayed on the 3rd floor of Buzz Lightyear, facing Andy's Room, on the wing closest to Cinema Hall (the main building). It was a great location and we could easily take the stairs to and from the room, which was the closest exit to the main building.
Yep, mugs are 10.99 (plus tax I think) and are good for free refills on sodas, hot cocoa, and I think coffee and tea, but since I never drank those I'm not sure.