If my son is tagged in a photo (who I am friends with) by one of his other Facebook friends that I am not a friend of and I like the photo, is there a way to get it to show up on my other Facebook friends' news feeds so they can see the photo of my son?
Yes, that does already happen. But Facebook doesn't show that on your wall unless someone comments or like it. You can find this by going to your timeline and checking your “Activity”. It’s best to share the picture on your wall—that is if your son's friend has made it shareable.