Welcmoe to disABILITIES.
For movies and shows, you will be able to stay in the wheelchair and sit in it during the show. For a few rides, you can choose to stay right in the wheelchair if you want (although most people who can get out of the wheelchair and get into a ride car choose to do that). Here's a
link to the DIS page about touring with mobility disabilities. The page lists attractions that have Mainstream Access (wheelchair users wait in line with everyone else) and also has a list of attractions where you can stay in the wheelchair for the entire time.
For rides where you need to transfer into a ride car, the wheelchair will almost always be able to come in line with you. For many rides, wheelchair or
scooter (
ecv) users wait in line, but then board at the exit. This helps because you generally have more room and less people while you board. Also, on most rides, the moving walkway can be slowed or even stopped to make it easier to board. (The only ride that can't be stopped is Peter Pan).
If you can walk a ways, they will have you park the chair and it will be in the same place when you get off. If you need the wheelchair brought up close to the ride (even onto the moving walkway), the CM will move the wheelchair out of the way for you and the chair will be waiting for you when you get off.
We have not had any problems with DD's wheelchair in almost 15 years of travel to WDW. We do leave her bag with clothes, etc on it, but I wouldn't leave anything valuable or irreplaceable on it when we're not there.
WDW park rental wheelchairs must get taken sometimes because they do have places where you can get replacements. It might be more a problem of "mistaken identity" though, since all the park rental wheelchairs look the same.