This is the waterjet I run, I've cut plenty of wood on it, mainly to use for fixturing weird parts, making templates, or just messing around. Mostly plywood and OSB. My experience is to not use the abrasive and do a water only cut and it should be fine. Haven't tried dimensioned lumber yet but it should be a similar situation.
As for the taper, it depends on the machine. Here we have an extra 2 axes on our Z axis, meaning it can rotate on the Z as well as pivot on the Z. So I can cut bevels, tapers, and all sorts of crazy geometry right from the waterjet. Being that we have this head assembly, there is a function in the software to compensate for the taper to make a perfectly 90 degree cut, provided no common line cutting is performed.