A:AnswerProbably but not an easy set up. But in what scenario would you want to record a program if this is for streaming services that most likely will be there later to watch multiples times?
A:AnswerWe have our Fire TV Stick 4k running through a receiver as well. No issues there. If your receiver handles 4k signals then you'll be able to stream 4k to your TV. If your receiver only handles 1080p signals then you'll be streaming 1080p.
I've read that Netflix (and maybe some others) require 4k @ 60 Hz. I don't know if it's true or not, but check your TV specs to be sure. The earlier generations of 4k TVs only handled 4k @ 30 Hz.