Network issue

[Efficient_Good_5784]

1. I wanted to test Jellyfin, so I made a host on one laptop (connected to the router via ethernet cable) and was unable to connect to it on my home wifi network. Upon realizing the problem, I disconnected ethernet cable and connected host laptop to the same wifi and it worked without issues. But if I wanted to keep my host connected to the router via cable and not the wifi, how would I go about setting it all up?

Is the router the same box that's providing the signal for the wifi network? Or are you connecting an external router to the box that provides wifi?

This could be an issue with different subnets. Make sure your router is set to connect devices to the same subnet as the one from the wifi network.
You can see if you're getting a different subnet on your laptop by checking what its assigned IP address is when connected with the ethernet cable, then when it's connected with wifi.

2. And going further I'm looking into building a NAS/home server. I really don't have much idea about networks, but I imagine that server would be also connected through cable and not wifi - so for example: how could I access Jellyfin hosted on the NAS from my PC connected only via ethernet cable and also other laptop connected to wifi?

This issue will be solved if you figure out what's wrong with your first question.
You'll be able to connect to the NAS on the same local network regardless of connection type.

3. Another level is accessing it from the outside (I travel a lot). From what I red so far it looks like I would need my own domain and some sort of reverse proxy setup. Most guides are centered around few other services, like tailscale/traefik/cloudflare, all these on docker containers, everything has to be set up, managed, etc... Seems like whole lot of complicated stuff - is there a simpler way to access Jellyfin library from outside of my networks?

The simplest would probably be Tailscale. Tailscale is a mesh VPN service which uses middle-men servers (Tailscale's servers) to connect all your devices through the internet. You can host your own VPN service too to ditch the middle-man, but this will be a bit more complicated.
With Tailscale, you don't need to make your own domain or reverse proxy. Tailscale will handle everything for you.

Just don't open a port directly to the internet. Doing so is the most simplest way to gain access outside of your local network, but this also is an easy way to get your network attacked from bad actors.