does chromecast have a bluetooth connection ability?
The third-generation model (2018) passed the FCC rated for Bluetooth capability, but it's currently not in use, and Google recommends the Bluetooth-less Chromecast Ultra for Stadia in any case.
You won't be able to play on a TV with a non-Stadia controller. The Stadia controller has a Wi-Fi chip and communicates directly with the game servers, so there's no extra hoops or hubs to jump through and latency is as low as possible.
However, you can use almost any third-party controller when playing through desktop Chrome or Chrome OS, or use any Bluetooth controller when playing on an Android device. However, as said, the Stadia controller includes dedicated hardware and talks directly to the Stadia backend, so it'll always be a superior experience to using a third-party controller.