Hi! We took our 2 children to Disney last summer. Our big one was 3 and our little one was 14 months old at the time, so he was just about the same age as your little one will be when you go.
I have no experience on planes with little ones, but I can help you with other things. We rented a double stroller from A Baby's Best Friend down in Orlando (they delivered it to our hotel and we returned it at our hotel for them to pick it up). We needed one for the two boys and we don't own one so we rented it. It was better than the ones at the parks because it was actually a Graco and perfect for the little guy when he napped in it. Much better than the hard plastic ones at the parks! We NEVER had a problem with the stroller being stolen! I felt very comfortable the whole time we were there. We even left the diaper bag in the basket at the bottom when going on rides and nothing was ever taken. And we are very cautious people (we're from Brooklyn, NY!). I had read many threads here about Disney and felt comfortable.
A couple of tips about using a stroller:
-cover it with a plastic poncho or something of the like in case of rain, when you are in an attraction. We went into Philharmagic and the stroller was soaked from pouring rain when we got out...however it did dry very quickly in the heat!
-put something on your stroller that will help you identify it quickly when you exit an attraction. We used a small handkerchief with bright print on it that I bought at Target for $1. Never had a problem finding out stroller...sometimes a CM will move it to make the row neater so this is essential!
-we never had a problem when navigating through a store with a double stroller, so you should be fine with that as well! So if your child is sleeping and you want to shop a little, go ahead and enjoy! I sure did!
Another good piece of advice with a toddler...try as much as possible to stick to their regular eating/sleeping schedule. It will prevent meltdowns. Remember, if you miss something you could always go back!
Have a wonderful trip!! You will love it and want to go back over and over again!