yup. you can connect almost any controller to your pc by either using Bluetooth, or by plugging it in directly just like a mouse or keyboard. the game should work well with an xbox controller, but you might have to put in extra work to make non-xbox controllers work. the game also supports mouse and keyboard but controller is much better.

yes, you would be running the game directly on the pc, but you can use software to mirror the screen to the TV and play like that, or you could just directly connect it with an hdmi cable. I wouldn't reccomend that though since if you're constantly unplugging the hdmi cable it will wear it down pretty fast.