These questions have been gone over so many times here .... what the heck, I can be brief.
I have a Dot connected to audio input of my stereo, and audio out from TV into a second audio input. So, yes, I do have to turn selector switch of stereo to hear one or the other. This is the way it is, and not convenient, although I have other Alexa devices nearby for giving commands to. If you want to use your Dot in the stereo at all times, dedicate it to that task, and just get another inexpensive Dot to place nearby.
Fire TV; The Amazon Fire TV app installed on phone or tablet can sub for the built in mic of newest remote. I also have the new Fire TV, and remote mic is not really that useful to use. Dot does not control Fire TV; better to use remote or Fire TV app. For them to work, the TV must be on if you want audio for weather, time, although it will work if TV is off and you ask Alexa to turn on/off lights as it can do, because there does not need to be audio for this task.