I don’t often dock my steam deck (have no controller so I’ll only use it as a media console) but when I do the only issue I’ve had was some applications starting in the wrong resolution when in gaming mode.

Otherwise I can’t say I could see the issues you’re having; I have no issues with sound over HDMI, my mouse and other Bluetooth devices stay connected in desktop and gaming mode.

Have you tried connecting another device (ideally also Linux) through the same dongle to check wether it outputs the sound through the Tv?