I sound to me that there something wrong with your setup.
First to answer to your questions:
a) The hearings aids shouldn’t shut off, only if you have activated the room off function (indicated by a red led in the bottom of the tv-controller).
b) Depending on your tv, adjusting the volume on the tv itself by using the tv’s remote control, can have no impact on the audio out connections only the tv speakers.
You should be able to hear both the transmitted signal from the tv-dex and the surrounding as long as the room off function
Here are a few hints to check the system. Connect only one audio cable to the tv base. Place the tv controller part the tv-base verify that it charges the controller green and red leds pulsating slowly. Remove the tv-controller from the base place it in very close proximity of your HAs and press the top button on the controller you should hear a sound (beep), press the bottom button this should mute the microphone on your hearing aids. If it doesn’t do this, it’s likely that the HA’s haven’t been matched/paired with the tv-dex. This is something your dispenser can help you with. Next hint, the volume can be turned very low on the tv-dex, so try to press the volume up(on the tv controller) button 10-20 times, if this increases only the “white noise” there is something wrong with the connection from the tv-base to your tv/amplifier. Try an another cable and check that you have connected it to an audio/line out connection on your tv, if you tv has more the one audio out try the other output.