How to teach students to ask Polya-type questions
Like most things, students need to be actually taught to do it. And "taught" doesn't mean having it explained to them, but rather practicing it via some sort of planned exercise or activity. 

It is an interesting idea, though. 

What I would do is to periodically hold a brainstorming session with students, assuming that the scale is reasonable. Take a problem that they haven't solved and ask for questions about it. Don't answer the questions, but ask for more questions - beyond the horizon questions - outrageous and maybe only potentially useful questions. Spend time on just gathering questions, making something of a game out of it. Note that I've done similar sorts of "gaming" activities with adult doctoral students, so don't assume that this just works with youngsters. 

Talk to them about what sorts of questions are generally useful. What is a smaller problem than this that we know how to solve? What is a related question? Where could we find information about this question? What do these words actually mean, individually and it conjunction? 

After you have a bunch of questions, try to have the students sort them in some way. Which are useful for this idea? Which are useful for extending the idea, or a related idea. 

If you practice this with them they will start to become more comfortable thinking that way, though you can probably expect more classroom interruptions in the normal course of things.