I just did some more testing and I found out the problems are related, as far as I can tell anyway.
The audio leakage slowly creeps in but only when I _use_ the mute button on the headset's wire.
The microphone actually works perfectly, conveying my voice very well (as my friends tell me).
However when I mute my mic (with the button on the wire), 'cause I want to watch a video or something, they start hearing the sound of the video after about 10-15 seconds, at first very quietly but then much louder.
If anyone has similar experiences, please share so we can figure this out.