I have a Briggs-R 6006 toilet with a #5164 Ballcock & Diverter. There is a water level adjuster screw above and coming out of the top of the vent cap which sits above the ballcock body. There is a knurled black Volume Control nob coming out of the side of the ballcock body.

Water is constantly flowing into the overflow tube. If I screw down the water level adjustment screw too far will that prevent water from flowing into the tank altogether, and what does the Volume Control nob actually do? Does it control how fast the water flows into the tank or does it have something to do with the water level in the tank?

That is the same toilet as the Case 1100. The black volume adjustment just determines how much of the water goes to refill the bowl. The screw on top adjusts the water level in the tank, so if you screw it down too far it will shut off the valve. More than likely you need the repair kit with a new stem and seat, among other things.