I think your cat will be thrilled to go with you. I'm headed to Orlando from New England, soon for a part time gig, so I will be parked and living at my job, which is at a house in suburbia.
I've been getting around without a toad and living, working, playing fulltime in my rig. I figured if I needed a car here and there, it would be cheaper to rent one now and then. So far I haven't rented one at all. My rig takes up less than 2 parking spots end to end, so pulling over for sightseeing or food shopping hasn't been a problem.
My little puppy dog travels with me. He is a joy to have around. I hope to add a cat to the mix, once the puppy is fully trained, he loves cats. I made a video of him playing with my friend's cat here http://dearmissmermaid.blogspot.com/2010/08/dear-miss-mermaid-makes-another-video.html
One thing about Florida and particularly the Orlando area is there are lots of toll roads. They generally move faster than the regular routes. You may want to get a prepaid toll transponder, as then you can whiz right by the toll booths without stopping. You can buy these at grocery stores, drug stores and other places. Read about it at http://www.SunPass.Com
Otherwise, keep a change purse up front full of quarters and one dollar bills. While most make change, I've encountered a few that require exact change (before I got the transponder.)
For cheap gas, check out http://www.gasbuddy.com/gb_gastemperaturemap.aspx
That shows a map of the USA and current gas prices, you can then check individual states and areas. As you can see from the map, you will want to top up your tank before you enter Florida.
You may want to get the cat a small cozy comfy cat bed that you can place somewhere securely while driving. My puppy dog loves his cat bed (he is only 5 pounds, the dog beds were way too big) and it either rides in the passenger seat or the club chair when I am driving. He generally stays in it 99% of the time, while I am driving.
I think you will be able to get into DisneyWorld's RV park, it's 750 acres. Once you get closer to an arrival date, you can make a reservation.
Otherwise stay somewhere else, then drive your RV to DisneyWorld for the day, have a reservation for that evening, so when you leave the park, you know where you are camping for the night.
Florida has tons of RV parks and campgrounds with prices all over the place.
My absolute favorite place to stay so far in my travels was Long Point Park on the Space coast of Florida near Melbourne Beach. I spent a week there and didn't want to leave. It's kind of remote, so it's best to show up with your fridge and pantry stocked up. It may not be everyone's cup of tea, but I just found the beauty outstanding and loved hiking the area.