13:58.09! Took a while to get sub 14 but faster execution on the 100 coin (4:46) helped me there. Here is the order.

1. Big Bob-omb on the Summit.
2. Footrace with Koopa the Quick.
3. 5 Silver Stars!
4. Big Bob-omb's Revenge.
5. Mario Wings to the Sky.
6. 100c.
7. Find the 8 Red Coins.
8. Behind Chain-Chomp's Gate. Yoshi did not enter the level with any caps, but rather picked them up at the designated area in the level. I'll try again soon!