I've been wanting to do exactly what you're doing here on my LAN for a while. I tried to do it on Friday using Zoraxy and managed to get Homarr running on server.local but couldn't get anything else running with a name (overseerr.server.local and server.local/overseerr just wouldn't work, although I did get a webpage on server.local/overseerr it wouldn't resolve properly).

Anyway as to your second point of getting a nameserver in Tailscale. While I haven't managed to get a nameserver in Tailscale I have managed to get apps running through Tailscale.

My app was Audiobookshelf. I wanted to be able to just turn on Tailscale on my phone and sync to Audiobookshelf and managed to do just that.

I already connected Audiobookshelf at home with it local IP.

I then spun up a Tailscale container in the docker host that Audiobookshelf was on, signed in to it on the Tailscale dash, then just added the Tailscale network in Docker to the Audiobookshelf docker container.

Now I can turn on Tailscale when I'm out of the house and open Audiobookshelf app and it connects to my.home server.

Meaning I don't need to remember the IP address and portz I set that up once in the Audiobookshelf app and connect to it at will.

I intend to have a go at attaching it to Syncthing next. I don't have much use case for Syncthing at present so it's a perfect app to experiment with. I intent to just attach the Tailscale network to my Syncthing container and just see if it connects. Then I'll try syncing my Keepass database to my host as an experiment from my phone.

In my head it should be that simple. If it is I'll just connect all my docker apps that way and spin up another Tailscale instance on my other VM that does my Arr, and I'll have outside access to everything.

Another point to give you for your quest: if you set up Heimdall and a Tailscale exit node, you can put all your self hosted apps in Heimdall for ease of access and then just hit that through your Tailnet. I have a shortcut on my phone home page. You can then just click the service you want in Heimdall and go to that service.

Edit: turned off the exit node I had running inside Home Assistant and now nothing works. Turns out it wasn't as easy to connect to the Tailnet as I thought, and I must have been hitting audiobookshelf through my Tailscale exit node after all. But that does mean that my final paragraph still stands, exit node plus a home page (Heimdall, Homarr) gives the same results, but without the nameservers.