So...alls I have to do to play Stadia outside of my home is take the Chromecast Dongle and my Stadia controller, hook up to another network(ethernet, WiFi, etc., etc.) and I can play on any TV even if it's not the original TV I hooked my Chromecast dongle to?
Technically yes you can do that, but you'll have to go through the process of changing the wifi networks (or plug the chromecast in to an ethernet jack that may be available). You can change the chromecast network in the Home app. Though I've never tried it I imagine you have to go through the controller setup process using the controller icon in the Stadia app again at this point.
The easier way to play anywhere right now is probably using the Chrome browser though as all you need is some form of PC running the Chrome browser.