Should security question answers be hashed?
Hashing it is a great idea, in my opinion. As no one really needs to know it other then the original user, so better to keep that more of a "Secret" from just prying eyes.
As far as the ability to change it, that is a great idea as well, and requiring him/her to enter their password to change it is another good security approach.
I think you are on the right track with your thinking.