Usually you have to make sure the other stream is off before it will work. It won't kick them as far as I know. I usually have to go turn one of them off in another room for it to work
Pretty sure YouTube doesn't work on FireTV. For some reason YouTube is not allowed on any of the Amazon products. That is unless something has changed recently.
Google/YouTube had a spat about 1.5 years ago with Amazon and at that time, YouTube would not allow its app to be on the firestick. About a month or so ago, the YouTube app came back to the firestick. However, the app for YouTube TV is still not on the firestick, so if you go with that service, you'd need to get a Roku device to play it on. I have both a firestick and Roku on my TV's and each one has their own pros and cons.