This isn't an answer but an argument that there isn't really a good answer. Having done a good amount of set theory and seen how you prove some of these statements to be independent, I tend to be rather skeptical about how reasonable these statements actually sound. Typically, while these statements sound like they're talking about some ordinary mathematical object, they aren't really, because their independence comes from very large and pathological objects that are far removed from your usual mathematical experience.
For example, in the Whitehead problem, you have to realize that while abelian groups sound very down-to-earth, uncountable abelian groups can have incredibly complicated structure. As a (fairly difficult!) exercise, you can prove that a countable product Z^N of copies of Z is not free, and in fact admits no homomorphism to Z besides the obvious finite sums of projections. On the other hand, the Whitehead problem has an affirmative answer if you restrict to countable groups, and this is something you could come up with if you thought about it for a week.
