Starting a sentence with "One question..." and then stating the question?

Solution 1:

With your first set of examples, either are correct. I believe It is quite reasonable construction. I think you're struggling in that the first part of the sentence sets up an indirect question:

One question you could research is whether female birds mature faster than their male counterparts.

As for the punctuation, this is a style issue. Either of your constructions would fit with The Chicago Manual of Style

In your bonus example, you are stating an indirect question so the punctuation is correct in the latter case.

One question you could research is what impact doing ten jumping jacks a day has on lifespan.

Still this is confused by using what because typically what is a direct question word. You're better off rewriting in this case:

One question you could research is the impact of doing ten jumping jacks a day on lifespan.