Once a movie I have rented from the AppStore has expired, and I wish to rent it again, do I have to download it again? Or can I just pay the additional charge and keep watching the file that's already on my device?
This probably depends on. If you just "recharge" within your rental window, the file is still on the device, and you probably don't have to download it again. However, once the rental period is expired, the Google Play Movies & TV App will delete the file on its next execution (see this thread on the DroidRZR forum), as it has "expired". Without this purge function, you could run out of space whith useless files occupying your disk, so I'd consider this normal behaviour.
This means: Re-rental after the rental window has expired will most likely make a re-download necessary -- while when still within your rental video, the app should simply keep the file (at least theoretically -- not sure whether the expiration date is somehow coded into the file name).