This is one of the those problems I kind of have to sit down and play around with. But in doing some research I found the below pin-outs for the audio plug. As you can see the plug with the microphone band is close to the base of the connector (Figure 1). Looking at the other diagram (Figure 2) you can see that the solid band (ground) runs almost half the length to the plastic base. So if you are only connecting headphones (figure 2) you are essentially grounding out the microphone input, which could be causing your problem. (common and ground are the same reference point, for this instance anyways)
(Figure 1)
(Figure 2) (standard 3.5mm audio plug for sound only, no microphone)
As far as the Bluetooth goes, I'm am not even close to being qualified to answer that question. I have so many issues with connectivity and communication with Bluetooth I end up doing in circles all the time. One question though, did your Bluetooth headphones also come with a wire? If they did then plug that wire into the headphones and the laptop and try the mic that way.
In any instance let me know how it goes and I'll keep trying to figure it out.
Also just a dumb question, you don't have a "mute" button on the wire that could be engaged?