While traveling I purchased a movie on my iPad from the App Store.
The purchase will be linked to your Apple ID, and can be (re-)downloaded on any of your devices with that Apple ID.
Redownload apps, music, movies, TV shows, and books from the App Store, iTunes Store, and Apple Books - Apple Support
How do I get a copy of the movie on my desktop?
Open iTunes, sign in, go to the Store, find the title. The title page will feature a download button.
or
Open iTunes, sign in, go to the Library, go to Movies, find the title. The movie poster will have a download-from-cloud icon below it.
Where will it be placed?
~/Music/iTunes/iTunes Media/Movies/[movie title]/[movie title.m4v]
Can I stream it from my desktop to a TV? Any help would be appreciated.
If you have an Apple TV streaming media box, or Apple TV App supported smart TV,
then you can play from there, or AirPlay to it from another device.