That's odd... What you're describing is a [u]Noise Gate[/u] effect, but that shouldn't be built-into a Bluetooth module. You might have a gate/squelch if the RF signal gets weak, but not if the audio signal gets week/quiet.
Maybe it's a feature to turn it off when there's no signal? (To save battery life.)
There's probably nothing you can do about it. I don't see any switches, pots, jumpers, or anything to "configure" or adjust the board and I assume there is no software configuration?
I noticed the board has some free spaces to solder SMD components, what are they for exactly?
Usually you'll see (small value) capacitors between the crystal terminals and ground. I don't know exactly why you usually need them, but if the board is working, this particular circuit works without them. ...Maybe they are installed when the manufacturer uses an alternate crystal.
The other part, I have no clue.
It's not unusual for a manufacturer to make multiple versions of a board, or multiple-different products using the same board (sometimes with different components).
Sometimes after the board is designed they discover a problem an a part has to be added (bummer) or they may discover that the board works with a part left-off. Actually, that's very common... Boards rarely work perfectly the first time and it's rare to fine a "Rev 1" or "Rev A" board in the hands of a customer.
If they discover an error on the board or they make a design change they'll usually use the existing boards if they can and they may, or may-not, re-design the boards for the next production run.