Do I need an STM lens for face detection live video with Canon? I bought a Canon EOS M to shoot video, mostly talking heads stuff so I simply set to the camera to AF Face tracking and had no problems.

I recently purchased the DSLR lens adapter and the cheap Canon f/1.8 50mm to use with the camera and the results are terrible. It just refuses to stay in focus for more than a second, even with someone sat still and not moving. It hunts none stop, creating a lot of off putting noise.

I understand that an STM lens will reduce the noise and the jerky movement, but do they affect AF as well? Is the lack of STM why I can't get solid AF?
The AF subsystem is part of the camera. It is not affected by the lens.
Face detection is also part of the camera.

STM only affects the speed of focus for a lens (assuming that the camera feeds it with correct information.)

So yes, STM lenses are better for video when coupled with a camera that has fast focus on its own. 

For your problem there are two solutions

1)Deactivate automatic focus and select a big aperture (like 11) so that everything is in focus
2)Deactivate automatic focus and manually focus the lens on your own.

All hollywood movies, tv broadcasts, commercials etc use manual focus.