Hello.
I am not at all familiar with parking in the UK, but I can assure you that there are "handicapped" spots in nearly every parking lot (car park?) in America. The passage of the Americans with Disabilities Act - ADA - is responsible for wider, well-marked parking spaces, curb cuts, ramps, elevators, etc. Obviously most of those accommodations are for people with physical disabilities including wheelchair users and others with limited mobility due to a variety of causes including respiratory illness. I am a mom to a daughter with high-functioning autism and I have never suspected that we would qualify for a disabled parking permit because she is not physically or neurologically impaired enough to keep us from walking longer distances. My concern for you is whether the state of Florida will recognize your permit. Most states have reciprocal agreements and recognize the permits of other states but I have no idea whether they would recognize the permit of another country. I would e-mail someone at the Florida department of transportation, or at the very least check with your car rental company. It would be awful to get a ticket, or worse, get towed, because of an innocent mistake!!!
Because I haven't spent much time in Orlando outside of visiting Disney, I can only offer educated guesses to your other questions.

So, Dissers, correct me if I am wrong...
Most attractions, malls, and hotels have parking lots, which will have handicapped parking. Just remember that Florida is chock full of senior citizens, many of whom have either permanent or temporary disabled parking permits. You may find all those special spots taken by seniors if you must make a trip to the grocery store.
I would not park
anywhere that is not a real parking spot even if the handicapped spots are full. Again, you will probably get ticketed or towed even with a permit.
I sincerely doubt Disney or anyone else will cut you a break on parking fees because of a permit. My mother-in-law had MS, hand-controls, and permanent disabled license plates, yet we were never offered free parking. Then again, we never inquired!

Maybe we should have!
Have a great vacation!!!