Does Netflix work over the Apple TV box (from an iPhone)

Apple TV's are NOT DVRs so they do not record, there are some apps (such as Hulu Pro) that does allow to record however the storage is NOT on the Apple TV it's in the Cloud hosted by Hulu. Also, if you bought an Apple TV 3 that is a VERY outdated device and you may not be able to use it. Please refer to: Identify your Apple TV model to determine what you own. If you can use it, great if not then please consider an Apple refurbished Apple TV. They come up for sale often on Certified Refurbished Products - Apple

Apple refurbished products are identical to a new device, have the same 1 year warranty as a new device, are eligible for AppleCare and can save lots of $$$.

A used Apple TV 4K 2nd generation would also be a good choice. It’s only 3 years old and has the new Siri Remote. Apple offers some refurbished : Refurbished Apple TV 4K 32GB (2nd Generation) - Apple.

if you get an appletv box then the best path would be just to install the netflix and disney+ apps on the appletv itself

Don’t get the Apple TV 3rd generation, you most likely won’t be able to sign in with your Apple ID because of the 2 factor authentication. Many users have this issue. Consider getting instead an Apple TV 4K.