If I share something on Facebook with my close friends list, can they share it to others? I'd like to share family photos with only my close friends, but I'm wondering -- can they in turn repost or share my posts publicly?
Not through facebook directly.
The idea of the "Custom" setting is so you can hide what you're posting from certain people, which is what you want and are doing. Because of that, facebook won't allow your friends on that custom list to share it with others because you have asked it not to by setting that post so that only certain people can see it.
Therefore if facebook allowed your friends to share it, they would be violating their privacy settings and not doing as you'e asked. Basically, if you set a post to custom because you don't want a certain person to see it, and then facebook allows that post to be shared, it could then be shared by one of your friends with the very person you didn't want to see the post in the first place. do you follow me? It's a bit confusing, I know.
That said, they could "Download Photo" and then upload it to their own album and re-share it - and this is in many ways much worse, because you would have no idea it was re-shared.