But if you want the true deals on Linux, I am working on that personally [1]. For clipboard sharing, it should be feasible on Linux if you have AirDrop compatible Wi-Fi hardware, but for iMessage integration, it's possible given Mac hardware identifiers. Apple requires attestations derived from Apple device identifiers to flag spam on iMessage. It would be great if Apple was also allowing deriving keys from the Apple account, so it can allow alternate device to borrow trust from their iPhone for example, but for now it does not seem possible (extracting identifiers once works though, and then you can use iMessage [2]). About Apple TV, idk I never seen this thing irl so I can't tell how that works but I'd guess it's replicable.
[0]: https://github.com/zackb/tether
[1]: notably in https://tangled.org/dadoum.zone/apple-crates but I also have more private crates implementing a kind of userland.