You set up a node on some other server and enable it to be an exit node, and can then access the Internet through that one. So any node on your net can be your VPN exit point at any time, if you want to juggle several, for example

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