Linux would be best for reliability. Nothing's better than untimely windows updates. Also, less power usage. If you do decide to go with Linux you should look into running jellyfin via docker. Makes it super portable and best of all makes it simple to run other docker services from the same machine (nextcloud, trillium, pihole).

You can set up duckdns for free domain names (with a docker service for dynamic IP) and use nginx as a reverse proxy to all your services (if you plan on streaming outside your house)

You probly could use tail scale. To acses jellyfin remotely. And automaticly sort /download things trough sonarr/radarr. If Windows is not a must I would probly recommend using ubuntu-kde on your laptop. There a script cald yams.media. That can do the install of all thing for you.

I have this exact use case too. I use Linux Mint with Jellyfin installed from the built in software hub. You cannot find it from the command line because it's a snap. The software app works and provides one click installation. Clicking on the jellyfin icon in your start menu brings up a browser tab that you can use to configure http access if you want. I use a raspberry pi running pihole and pivpn for a VPN and the wireguard app on my phone to connect to that VPN. This works on my cellphone over the network and allows me to access the jellyfin media via the jellyfin app. I think it works nicely. I have jellyfin media player on the laptops and the streaming works on the lan. I will caution that I haven't been able to easily set up wireguard on the laptops. I use phone to stream, so this might just be a motivation problem on my part.

Internal to your LAN/WLAN Jellyfin will work right out of the box it collects content from whatever folders you tell it too. Phone, smart TV, etc. You might need local clients on some devices.

Outside your local network it gets complicated, has security concerns. Certainly doable, but outside my expertise. If you want to serve outside your local network, PLEX is a similar, close source, software that can take care of the web for you.

It looks like setting up casting is possible, but a giant pain in the ass according to this guide. I would see if your TV has a Jellyfin app (they have apps for Android TV, FireTV, LG WebOS, and Apple TV), otherwise you might need to just buy a cheap Roku.

In terms of watching outside your house, that will probably be difficult. Because we ran out of IP addresses, most ISPs use NAT which basically bundles a bunch of different users behind a single IP address, making port-forwarding impossible unless you pay extra for a static IP. You'll have to figure out whether this is the case for Cox.