Read this section on setting up forwarding on Linux. You’ll want to do this on the host that is running docker and it should carry down into the container itself.
Yes, you can run Tailscale in a container. You could create a second VLAN, attach it to your hosts interface, add a macvlan docker interface to the container and put it directly on your network.
If you have concerns about the software running on your host I would recommend getting a dedicated piece of hardware instead (rpi, zimaboard, etc).
How paranoid are you wanting to be? You can either go Headscale, or Tailnet Lock (my preference) to give your self some peace of mind. It completely depends on your threat model, which you didn't mention.
The latest Duck Tales, that might be the first episode.