1. Does Apple TV stream (I don't want to save my movie files to my laptop). If it can stream how can I put the movies in the Cloud?
Not sure what “my movie files” consists of. Is this about purchases from the iTunes Store? Then those are available to stream from Apple’s servers without a need to download first. No action needed.
Is this about personal home videos? Then you can store those in e.g. your iCloud space/iCloud Photos. If you have a lot of such videos, then you may need to upgrade your iCloud plan (5-50-200-2000 GB). It is good practice to also have a local backup of things that are too important to risk losing.
For other kinds of videos, ask yourself if you have permission from the content creator/rights holder.
2. Does it synchronize between MacBook Pro, iPhone and iPad? Do I have to tell it to do that?
Content on remote servers (cloud) is available equally to all your devices.
3. iPhone has an extra movie on it not on other devices. How do I rescue it?
Syncing media files to iPhone (TV app library/Music app library) is one way. I do not know of a method to sync back to a computer; it is probably officially unsupported. You should have a kept the original on a computer. Only videos in the Photo Library can be synced (transferred) to a computer.
4. iPad has many movies, shown under recommendations, which I want on all devices but it seems I can only keep on iPad or delete.
Which recommendations is that, where do you see that? The ‘Genius’ recommendations in the iTunes Store app are an account feature, based on past purchases and optionally some library content. That shouldn’t be specific to a device.
Do all these devices have the same Apple ID account?
What is the version OS on the iPad?
