To answer more directly the original question.
Aircraft do have vertical cg limitations, but they are generally a "design consideration" only, with the intent being that the eventual operators don't have to worry about vertical (and, indeed, also lateral) cg loading considerations, because the OEM has considered a representative cg envelope in all three axes (plus associated inertia effects as appropriate) in developing both the airframe and component loads, and assessing the aircraft handling characteristics.
Certainly on a small aircraft carrying external loads representing a significant proportion of the overall mass, the resultant vertical cg variation was not insignificant and was considered as part of the conditions for each clearance.
For larger aircraft I'd expect the norm to be, as for lateral cg, to consider various nominal and worst case loading scenarios, and use them to create a design envelope. Generally speaking the effects are both smaller than the longitudinal case and also represent smaller variability, so there's little risk of an abnormal configuration taking you to an unsafe condition.
To address the question of low-slung engines and aircraft dynamics, cases that have to be considered include both the basic trim requirements of 25.161 (or equivalent) which cover a range of power settings (also implied by various other regs), plus dynamic cases including both the GA case mentioned and, quite important, stall recovery - you have to assume the pilots will be applying power to accelerate at the same time as either they or the SPS/FBW system is trying to lower the nose; the engines will therefore negate some of your recovery control power, and this must be considered in, for example, tail/elevator requirements.