Thank you ss332 for your analytical approach! Sorry for the delayed answers,
Answer 1: Yes, all my Bluetooth headsets, including the LG HBS-800, are easily paired and connected to my Samsung S4 phone and to my Asus Nexus tablet.
Answer 2: My LG TV was in the discovery mode not visible for my phone and tablet. The Samsung TV of my neighbor was visible and also his printer.
This result brings me to the idea that there is probably no Bluetooth signal at all coming from my LG TV!
I downloaded the LG Wireless Sound Sync manual pages and found:
Manual 1: Check whether LG HBS / LG BTS1 is used in the name of the headphone.
My LG HBS-800 headphone is visible on my Tablet as HBS-800, without the name LG, but I suppose that the filter uses HBS-800.
Manual 2: For some models you need to connect a Bluetooth-dongle to use LG Sound Sync !! ( Accessory of the TV or separately sold.)
Question: I would like to know which Bluetooth specification or which Bluetooth dongle model I need ?