Enough people have already told you, Disney allows you to check your bags the day of your checkout, and you can pick them up when you're ready to head out that day. So don't worry about that.
As far as a place for a kid (OR an adult) to take a nap is concerned, in the MK, there's a Kodak pavilion right next to the entrance, before you even start heading down Main Street (next to Tony's Town Square resturant). In the back, there's a scaled-down theater that plays Disney cartoons. There's also some cutouts of classic Disney animation in the back you can use as a photo oppotunity (for example, you can make it look like you're in the TV the 101 Dalmations are watching). The lighting is dim but not dark. The soundtrack is not too loud. As long as there aren't other loud people in there with you, it's a great place to relax, watch some cartoons, or nap. I've never seen it so busy that you can't find a seat, and I've never been asked to leave because I dozed off.
Otherwise, every resort has plenty of lounging areas, and I doubt anyone would complain if you sat quietly with your child while he napped. Whatever park you go to, once he starts getting cranky, head to one of the nearest resorts. One of you can get some coffee and snacks while the other sits with the kid. I'd also suggest going to the Animal Kingdom Lodge. There are areas where you can view the animals they keep in their private savannahs, and the decor is actually very soothing.