I’ve got both a fire TV stick 4k and acorresponding Mi tv stick, both mainly for jellyfin, and would recommend the latter. Basically same price, both work fine for the purpose but fire has their ads on the home screen whereas Mi is more clean and has the Google play store, making some things easier to install. The Mi apps you can just disable.

You can try having the same hostname resolve on the lan to the local IP and only connect using that first part of the addess, having dns suffixes do the heavy lifting

Yeah you should be able to set a custom dns for your whole tailnet I think, going by memory. I know I’ve had some issues but just try it, it’s quite fast to set up :)

Yes that’s what you can use the tailscale subnet routing for - one some machine on the lan, that joins the tailnet, you configure it up to route to local subnets