It will be fine. If two computers are on the same lan, they will connect directly using your local network. If they are on two different networks, they will reroute through their network.
Will I be able to just launch Tailscale as a part of my normal startup? Can Tailscale start 'invisibly' that is without any interaction from me as I turn on and login?
I think so? I don’t use it on windows so I don’t know how it works there. All other places I used it you can set it up as a daemon or as a startup app.
Yes, Tailscale starts up with Windows and doesn't require any interaction.
One caveat, DLNA/casting/KDE Connect don't work when Tailscale is active. Seems to be a limitation with multicasting not playing well with VPN-s in general.
Yes, it has a start up service in windows and linux.
I can tell you the android app doesn't start automatically, you need to access it each time you restart your device. I don't know for iOS.