To me, the main difference is that relationship is broader than relation:
- both can mean “the way in which two or more concepts, objects, or people are connected”;
- in addition, relationship can mean “the state of being connected”.
So, in “She has a close relationship with her daughter”, because you're talking about the fact a particular existing connection, I wouldn't use “relation”.
In your second example, if “the relationship between poor housing and health problems” is merely an academic study of the possible correlation, then “relation” would fit well.
In the third, again, the “special relationship” is a particular, existing connection, so “relation” wouldn't fit the bill. In addition, “special relationship” is a well established phrase since post-World War II times.