I've used ECVs a few times now, the first time I rented it for 2 days in the parks, and the last 2 times I rented them for the week from ScooterBug. I can't imagine how anyone would steal one unless you left your key in it. The cast members will move them. That's almost a definite, sometimes they rearrange to fit more, and sometimes they have a designated area that they move it to by the exit.
Here's my experience with rides and lines...
Magic Kingdom
Tomorrowland - you have to park the ECV by the exit ramp of People Mover for all the rides. (I can't say for sure about Space Mountain because I have a hard time getting off the ride so don't go on it anymore)
Peter Pan's Flight & Winnie the Pooh - you have to park it with the strollers, but its hard to find a spot
It's a Small World - go in through the exit, and the attendant will show you where to park, and you will mostly skip the line
Pirates - you have to park the ECV by the side of the gift shop
Barnstormer - ECVs can go through line
Haunted Mansion - ECVs can go through line
Little Mermaid - ECVs can go through line
Jungle Cruise - have to park ECV somewhere, I think I just found a place near a fence
Epcot
There is pretty much no real ECV parking areas in Epcot.
The Land
People park ECVs in the Land by the railing when you first go in. But you can take them downstairs in the elevator.
Soarin - ECVs can go through line
Living with the Land - ECVs can go through line
Nemo & the Seas - ECVs can go through line, but it is the hardest lines to get through with all the turns. When you get on the ride they will move the ECV to the area with the fish, and you can leave it there while you walk around the seas, or hop back on it.
Figment - ECVs can go through line
Ellen's Energy Adventure - have to leave the ECV parked outside somewhere
Spaceship Earth - have to leave the ECV parked outside somewhere
Frozen Ever After - ECVs can go through line
Grant Fiesta - have to leave the ECV parked somewhere, perhaps outside the Mexico building, I forget where I parked for that
The American Adventure - you and everyone else with and ECV or wheelchair will be loaded up a few at a time into an elevator to go up to the 2nd floor. If you can find a place to park and walk, I think I'd recommend doing that.
Canada and China's 360 degree movies - you can stay on the ECV the whole time
Hollywood Studios
Tower of Terror - ECV's have to be parked somewhere, and I don't know where.
Great Movie Ride - ECVs can go through line
Star Tours - ECVs have to be parked somewhere, I think by the wall across from the entrance
Toy Storia Mania - ECVs can go through line, though I think there are some tight turns
Any of the shows - ECVs can go through the line.
Animal Kingdom
Kilimanjaro Safari - ECVs can go through the line
Any of the shows - ECVs can go through the line
Wildlife Express Train - ECVs can go on the train
Maharajah Jungle Trek & Gorilla Falls Exploration Trail - ECVs can go through the trails
I base my Magic Kindgom Fast Pass selections on whether or not the ECV can go through the line or have to be parked before getting in line. Pretty much always Pirates, Pooh and Peter Pan. I think most of the lines that the ECVs aren't allowed in will let you transfer to a wheelchair for the line, but I travel solo so have no one to push me, and don't know how to use one myself.
Here's some pics from my last trip, perfecting the Disney ECV adventure...
That's a phone mount for bikes on the handlebars. I used pipecleaners (they have some new crafty name for them now) to hold up my water bottles and battery pack for my phone.
I used hooks that go on the back of car headrests for my backpack, and a two-piece combo lock. I mostly left it unlocked so I could easily take the backpack off. But I used it as a theft deterrent since I can't see the back of the ECV while riding it, and someone could easily grab the backpack off the hooks without it. Also as support for the bag since the hook came off a few times.
This particular ECV is the only I know of that has a headrest like this, and I purposely order it so I have a place for my bag. On the previous trip I just wrapped the backpack handles around the headrest bars.