Question about Ugreen CM403

Yes

Yes, you can, as long as the source (TV) has a 3.5 mm Audio out (Headphone or similar).
Pairing is between transmitter and speaker and works on speakerside as usual, and the transmitter you have to set the slider switch to sender and then press the button ca 3s to pair. (Check also manual)

But I would not recommend it in your setting, because the sound will have delay like all bluetooth connections. And the device need to be turned on every time (auto Off after several minutes without input)

Yes, pairing works with a simple click on the on/off button. The sound delay mentioned by Peterrothe is not very noticeable. It works fine in my home.