Is there a book/survey-paper outlining language class hierarchies, closure properties, etc
Not standard topics, no. And sorry, I have no general overview.

However, I would have a look at the PhD thesis of Klaus Reinhardt for at least a picture of the various families that live in this area. See page 64 for a diagram of the zoo. Motivated by Petri Nets with inhibitor arcs Reinhardt studies priority multicounters with restrictions on when to do zero tests. Non-trivial. 

By the way, your last example question was discussed in this forum by user Sam Jones. Another Ibarra reference.