This is confusing as live view Quick AF is using the same focus system as looking through the viewfinder, so does it achieve accurate focus lock on Quick AF, or just go through the motions and miss? I can't see how live view Quick AF could work yet viewfinder AF and even manual through he viewfinder could miss!
Another basic possibly stupid question but is the lens fully engaged and locked, or could it have stopped short of locking somehow, affecting Lens body communication?
Have you checked the secondary mirror, perhaps the flipping down of the main mirror for live view Quick AF is forcing it in to the correct location where a static start is allowing it to be out of line?
When you get to the bottom of this do tell us as it is an interesting situation that most of us will probably never experience.
2 & 3) Manually focusing works through live view and the AF even works in live view in Live AF and Quick AF.