after call it turns on music player

Is this happening while the device is connected to a vehicle, or is this just from holding the device in your hand?

While you probably know this, it shouldn't be happening, however have you tried resetting the headset? You may have to check with the manufacturer to find out what the procedure is for that. I've got a BlueAnt in-ear device and I do not experience that behavior in my 5S, or 4S that I had before that.

If you do not have an answer, do not exhibit ignorance and in any way indicate that the other party is to blame.

This is becoming quickly just like Microsoft. Now, something worked, and now it does not work. That means that something has been done in the Apple provided software or hardware.
Here the answer is very simple: There is an assumption that has been made by some jerk that people that use Bluetooth headsets do so to listen to music. And Apple then assumes that all music is to be played by iTunes.

Now: Tell us where in the preference .plist we can remove this once and for all. Because:

When I turn on a BT device for my phone, this searches for connections - on Android, iPhones, Macs and iPads lying around. And the latest is that it can be connected to multiple hosts, meaning that unless I reach safety quickly, I have 5 bands playing various music and it is an utter nuisance.

Now, there is no difference between a wireless headset to one that has a plug and inserted in the hole in the side, it just is more convenient to drop the cord. So, remove all special features, bells and whistles - and focus on getting things to work.

This is an American problem - people in the US have not used cordless headsets.