We used the small net pockets for drinks as well (this was on a double rental stroller, I can't comment on the singles). It worked fine as long as your drink wasn't full full. Of course, drinks don't stay FULL for very long when you are at disney in the summer, lol!
Also, while the strollers are hard plastic, my kids (3yr and 5 yr at the time) didn't seem to care. They were just happy to have a seat and a place to nap at times. If you think you kids will be bothered by it, the bathmat seems like a great idea! We did bring a clear shower curtain liner to use as a rain cover since we were there during the rainy summer season. I brought clips that I took off the bottom of helium balloons from DS birthday to secure it (they were plastic clips with a metal spring). We just had to be careful that the shower curtain didn't get caught under the wheels. We also used the clips to attach the park map to the top of the stroller so it was open and handy when we needed it.