You raise two issues:
(1) There is potentially some overlap between this proposed Stack Exchange and the proposed Eduction Stack Exchange.
(2) We should be careful about questions that are "too soft".
For issue (1), I'm not sure this is something we have to worry about unless both of these Stack Exchanges make it to beta. Even in that case I would lean towards being flexible--any question here is being asked in the context of teaching math, and I wouldn't want to migrate a question unless that context clearly had no effect on possible answers. For example, the question
What are the best teaching methods to be a good tutor?
is clearly being asked in the context of math tutoring, which is pedagogically very different from, say, writing tutoring. (I do have some unrelated issues with this question — it seems extremely broad, and it would be better if the questioner could narrow it down a bit.)
How do I deal with one very bright and vocal student who shouts out the answer to every question?
is closer to being perhaps better suited for the Education SE, but even in this case I think this is often a challenge that's specific to math classes. The rhythm of a math class tends to be very different from classes in other subjects — even classes in other STEM fields — in that the whole class often needs to stop and think about a problem for a few minutes. If one student shouts out the answer early, it can short citcuit this process, and the other members of the class don't get the necessary problem-solving experience.
This is a reasonable concern, but I don't think the questions you mention are any softer than, say, a typical question on the Academia Stack Exchange. The question
What middle and high school topics should be taught in mathematics?
is again quite broad, but if it were narrowed down a bit I think it could be an excellent question.
At what stage of development should students be first exposed to the notion of a proof?
How do I approach teaching a student who needs math but does not enjoy it?
What role should proofs play in a first calculus course?
seem to me to be reasonably clear and specific, and there might very well be research in the literature about all three of these. It is true that different teachers or professors might have different opinions about these questions — which makes them a bit subjective — but there is still a lot of value in learning the range of opinions that exist, as well as the arguments behind them.
At worst, questions like this should be made "community wiki", and I think we have enough specific, non-soft questions that we fit reasonably well into the Stack Exchange format.