I can't tell you I've tested every headset out there, but it is standard so should work (always has for me). The one exception may be wired Apple headsets that use more than the basic single mic and button arrangement, as they use a proprietary system and will likely struggle.
I have tried this on about 8 or 9 different headsets. All have worked.
Yes, when you hold down the call button Cortana launches (even if phone is in standby) and you can speak to it through your headset mic. Seeing the results, however, often requires that you find your phone and look at it so not as useful as one would hope.