I have no experience with the VexPro Ball Shifter, so I won't go over that.
I have had plenty of experience with misaligned axles and sprockets, wobbly bearings, etc.
WCP's gearbox is designed such that if it's mounted on the centerline of a 2x1" rail, it's really hard to mess it up, no matter how much you over-think the drive train design. The mount holes of the WCP block for the output shaft have to be fairly precise, but the outer two mount holes have some tolerance.
When combined with the
VersaBlock kit sliding bearing block, you can make a precision WCD drive train without the frustrating time wasting mumblings that come with messing up a non-DRO manually milled part by 0.03". Note that you want the WCP gearbox to be on the centerline of the rail, as the VersaBlocks have the "wheel drop" built-in to their design - the wheels mounted to these are raised by 0.06" (which, by the way, means you can place a precision-aligned raised wheel using just a hole saw).
As for chain vs belt - the WCP gearboxes have appropriate space for chain, but I don't know if they can do belts without some modification. The WCP gearboxes also come with wave washers to dampen axial vibrations which cause chains to pop off under a variety of loads. CAD it out first if you have your heart set on belts.