Hi, Terry: In each example, "Something" is being posmodified by a phrase that has been moved to the right:
1) something to do with my daughter
2) something having to do with my daughter
3) something requiring my attention
4) something that requires my attention
As you can see, you are right that "that requires my attention" is a relative clause in (4). "Something" is likewise modified by relative clauses in (1), (2), and (3), but they are reduced, non-finite relative clauses.
As to the extraposition, it works the best in (4) and is OK in (3). Sentence (2) isn't terrible, but (1) is. I recommend moving the phrases so that they follow the word they modify ("something"), as illustrated above. Or you could use two sentences:
- Something just came up. It has to do with my daughter.