You will be able to stay in the wheelchair for all shows and movies; there are spots for parking the wheelchair to view the show. Here is a picture of the seating area for Fantasmic. The wheelchair parking spots are outlined in blue. There is a spot next to each wheelchair spot for someone to sit; the rest of their party can sit in front.
SOme attractions also have a wheelchair accessible ride car/boat. Here is the picture of the wheelchair accessible boat from Jungle Cruise:
Here is the wheelchair accessible boat from Small World:
and the ride car from Imagination:
A few of the wheelchair cars have a bump or something else on the floor that may prevent your wheelchair from going on. Here is a picture of the accessible ride car on Triceraptop Spin, which shows that situation. Aladdin's Magic Carpets has the same situation.
My DD has an Invacare TDX power wheelchair also. We have not taken it to WDW, but I could see that it would not have enough clearance to go on this ride car. If you are able to transfer to a regular seat, you would be able to park your wheelchair and go on a regular ride car. Another option would be to move to a manual wheelchair at that attraction and you could use that in the accessible ride car.
Some other attractions have a wheelchair boat/car where a sharp turn is required, but the TDX wheelchairs can turn very sharply. THis is the boat for Living with the Land:
and the car for The Great Movie Ride:
The ride tram for the Backstage Tour at the Studio and for the Safari at AK are almost the same in terms of getting in and out:
If you look in the disABILITIES FAQs thread, there are some links in post #3 that will help you - there is a Trip report from eternaldisneyfan that talks about her trip with a wheelchair to WDW.
the chair will be placed at or near the exit where you get off the ride. they will not specifically assign somebody to guard it though.
There are a few attractions where they will move the chair for you from where you get on to the exit.
For most attractions where you enter on one side of a track and exit on the other, wheelchair users are sent to the exit to board (usually, you will wait in the usual line for a while, but are re-routed just before the regular line gets to boarding).
they are usually in an out of the way are where there is no general access from like crowds. if you are worried, if you have some way to lock the wheels or secure it from being started, you may want to try that

Usually, the ride will be at the exit and since most people are focused on getting out, they don't even notice things like wheelchairs parked there.
Once when I went on Haunted Mansion with DH and DD. When the ride was over, I got off. Unfortunatly , the CM forgot to stop the moving walkway for DH and DD to get off, so they continued on for one more ride. I stood at the exit and waited for them. I would bet that no one noticed me at all.
PS - I am going to move this thread to the disABILITIES BOARD. That is much more aimed at WDW trips and you will get more answers there.