Simple stochastic games on pushdown automata zoo

I will try to answer some suitable superset of the following questions: What the hell are simple stochastic games? Why should someone want to play it on pushdown automata? What are the natural habitats of the animals in the zoo -- one-counter machines, stackless pushdowns, etc.? Is it only the taiga of modelling and verification, or do they appear also in the queuing theory forest, language processing desert, and game theory gambling mountains and valleys? Which of them are inevitably dangerous (undecidable), and how long does it take to tame the rest (complexity)?
Everyone is invited to ask their own questions as well.