Okay, let's clarify some things:
1) Dale is correct on both counts. Yes, it must be a CFI rated to train for a Private Pilot ticket. And yes, finding a CFI that holds only a Sport Pilot license is about as rare as the Yeti in Kansas. I wonder if there actually are any Sport Pilot only CFI's at all. It's so rare I didn't mention it.
2) As he said, the check ride must be in an LSA compliant aircraft. Training, however, can be in anything. I had two hours in a C172, for example, and it counted towards my total. Likewise, my biannual flight reviews were in aircraft too heavy to be LSA (a decathalon and a C172).
3) Renter's insurance! It's pretty cheap and when one is flying solo they'd be covered, as at that point one is renting the aircraft. And yes, they sell it to student pilots.