Connect LG HT to TV with HDMI. If both TV and LG have ARC turn it on in TV menu. This will send audio from the TV to the HT. If they don't have ARC then you will need to connect the TV audio out to the LG, turn it on in the TV menu and maybe turn the TV speakers off to get it to work.
Connect the game to the TV. Game sound goes through the TV to the HT. Select input on HT you used (ARC or optical)
You can't play a disc on the HT and use a mic through the HT speakers. You will need a separate karaoke disc player that has mic inputs.
That would connect to the TV and the sound would go into the HT through the TV.