No, you can't use your own domain, this specifically works with the ts.net subdomains. You could use it to host your service on "node.tailnet.ts.net", if you're ok with that naming. Otherwise, you'll need to set up a VPS with tailscale on it, and your own domain, and have it proxy similar to how Funnel works.
I assume there are going to be some egress charges for Funnel. Unlike normal tailscale, Funnel has a high bandwidth component to it for hosting high bandwidth services.