Nursery rhyme trio for BEARS refers, I believe, to "Goldilocks and the Three Bears," which is not a rhyme.
Subway rider's aid for STRAP is about fifty years too late, and while riders still may be referred to as straphangers, the clue should indicate the past.
To improve some rather prosaic clues, I would have changed:
One with a code name, perhaps for SPY to
Bond or Smart.
Fireside stack for WOOD to
Driver.
Once around the track for LAP to
Dog seat.
Paleozoic ___ for ERA to
Flapper or Rock.
