Studying cellular automata with methods from communication complexity appears to be a promising approach. In the past, interesting connections between communication complexity and intrinsic universality in cellular automata were shown. One of the last extensions of this theory was its generalization to various "communication problems'', or "questions'' one might ask about the dynamics of cellular automata. In this article, we aim at structuring these problems, and find what makes them interesting for the study of intrinsic universality and quasi-orders induced by simulation relations.