You have two modes of liveview...Tripod or Handheld. The shutter button or a remote electronic shutter trigger does not function at all while in tripod mode liveview, and will only function when fully depressed in handheld mode. In both modes it will autofocus only right before making the exposure and not while simply viewing.
I don't use handheld mode as I am always tethered and on a tripod which has some other differences in the way it focuses so I will have to look into it more. Not sure exactly how liveview functions while in continuous focus mode.
P.S. It didn't used to be this way. At least with camera control pro 2, for some reason they took away that functionality some time ago.