We have stayed at Coronado Springs and the Animal Kingdom Lodge during our stays at Disney. Coronado Springs was nice and we loved the Animal Kingdom Lodge (our room had a queen-sized bed with a set of bunk beds and we woke up to see zebra, giraffe and antelope outside our window). It just depends on your budget. I would highly recommend you stay at a Disney property. All your transportation is provided to and from all the attractions and is so easy. Go to the Disney World website and browse through each of the resorts. They have places to fit everyone's budget. Height restrictions are also provided on the website. My kids are a bit older so we didn't have any issues but I was very surprised at the number of rides that little kids are able to go on. Disney works very hard to make things accesible for the little ones. Also, check out MouseSavers.com for special deals. There's also a Mom's Disney forum where you can ask Disney moms for advice (sorry, I don't remember the link). Regarding the stollers, remember you'll be walking around the park which can get quite crowded at times so trying to navigate with a double-wide stroller might get a little tricky.
Don't forget to get a PhotoPass as soon as you get there. There should be a kiosk in the resort that can get you set up. Take tons of photos with the park photographers and have them scan it to your PhotoPass. You don't have to buy anything, but in case they capture that 'Disney vacation moment' you don't want to miss it. Disney has some great items that you can have your photos printed on so check them out online before you go. Their story-book is really cool and not too overpriced considering all that you can do with it. I took all our PhotoPass photos, used Disney's photo shop tools online and had them all burned to a Disney photo CD. I then took that CD to Walgreen's and had them printed out at 20cents a piece. The CD seemed expensive but once I figured out what I'd spend buying the photos from Disney, it was way cheaper to just by the images and then print my own.
Make sure they know you're celebrating your birthday! They'll give you a pin, stick it to a ball hat and wear it in the park. It's amazing the special attention you'll get!
Have fun!!!