If a friend posts a picture of me on Facebook with privacy settings so only certain of his friends can see it, can I use the "Share" functionality to share it to my friends who are not his friends?

Right now, if you are tagged in Facebook content your friends will be able to see the picture/post - even if they are not friends with the person on whose account the content resides. You don't even need to "Share" it. (In other words, if your friend tags you in a photo he posts, your friends can see it straightaway.) If you are not tagged then the privacy settings will be inherited. So if you "Share" it only those who could see the original content will be able to see the shared content. If the original photo was public this will not be a problem but otherwise it can get annoying. If you don't want to be tagged for some reason (or if you want to share something that is not directly related to you) the best thing to do is copy the content and repost it using your account instead of sharing it. Incidentally this is why in Facebook groups people often post content that other group members cannot see - due to privacy settings of the original content. You will see: https://www.facebook.com/groups/sgskeptics/

You could also probably share the URL of the specific image (not the image page), although I haven't tried "sharing" it through Facebook that way. It works when you, say, email or IM the person the URL. That's not exactly the most ethical (your friend did only designate the photo to be seen by a subset of people) or efficient (if you want to share a lot of photos) way to do it though.

The creator of an album is given a public URL which allows them to share the album with anyone. You can ask them to provide it for you and you can pass it along.