How do I find out what's the minimum version of each dependency my module will still run with?

This has already been said by several people that you should give considerably more weight to than I, but when I read your question, this was my first reaction. The difference is that I'll tell you why. My answer to the above question is:

Don't!

That is, if you don't know what to do, do nothing. When you know action is required, you'll probably also know what action is required. But speculating about what action might be required, before you have evidence it actually is, is counter productive. In English. A total waste of time.