iTunes VIDEO
shouldn't automatically download to iOS devices.
https://support.apple.com/en-us/HT202180
View attachment 728408
I just checked TV app, and while all my purchases are front and center they aren't downloaded to any of them except the ones I intentionally downloaded them too.
However if you internet is fast enough as soon as you click the purchased video icon it starts streaming. However it still requires me to download it manually video the cloud button (like Apple Music).
View attachment 728409
Are you sure you aren't just streaming it?
And yes, purchases will appear on all your devices that are using the same AppleID (AppleTV, iPhone, iPad, iTunes on Mac/PC). So yes if someone else is using one of your devices or their own device that is using the your AppleID (I recommend avoiding that) or is using family sharing (preferable) they can watch the movies you purchased on another device.