@PawanAurora: above, Little Addition is linked to your example, and Little Addition 2 is linked to Guenter Rote's answer.

@GuenterRote: in dimension three you may wish to consider fullerenes (liga.ens.fr/~deza/Sem-FullCCirmVirusSpFull/FFullereneConf.p‌​df) as an example of polytopes with many vertices of degree 3 sufficiently far from each other. I think it works, when you cut a number of vertices of a fullerene by triangles, since fullerenes have 5- and 6-gonal faces only. The number of vertices of a fullerene is generally unbounded.

@PawanAurora: I edited my (partially wrong) answer. I will see if I can say anything more now, when you clarified the situation with that nice example.

Thank you! Now I see where I was wrong about the case when you consider facet defining hyperplanes instead of faces: F_P definitely has a new hyperplane, but when it intersects those already in F_Q, it may happen that no new vertices appear (thus, the new hyperplane has to go through some "old" vertices and may chop off some other "old" vertices.

There are no compact right-angled polytopes in dimensions >= 5 according to Potyagailo and Vinberg. Thus, you cannot expect closed manifolds with right-angled fundamental domains (which [domains] are necessarily compact) in dimensions >= 5. However, you still may construct manifolds with cusp out of right-angled polytopes, which are known to exist up to dimension 8 (Potyagailo and Vinberg, again). The upper bound for their dimension is <=12, however, there are no examples in dimension 9 to 12.

@few_reps: I would be happy to see your paper on arXiv soon. Reading it will be a very interesting continuation of what we discussed before. I'm interested if your method can generate Coxeter or rational-of-pi angled polytopes in dimension four.