Is it possible to download on-demand movies I have purchased to my tablets so I can watch them without an internet connection?
Only if they came with a digitial copy (i.e. UV or Disney) otherwise the answer is no.
Depends on the form of download. Some play with you devices videoplayer. For those check your download folder.
Most of mine are thru VUDU and you just go to the VUDU app and look for myvudu, downloads. With VUDU if you are offline (e.g. airplane mode) it launchs directly to the downloads. Haven't tried downloading any of my ITUNE movies, but I know the audio plays similarly.