In the beginning of the level take a path on top of the trees that leads directly to the key. Tonguing the key (see IL) saves about a half a second. Just flutter into the ceiling, then tongue when you hit the ceiling and immediately hold left. You may need to tongue the key a second time to bring it all the way to Yoshi.
There are two ways to skip pushing the chomp rock through this level. The top route is easier, but about 2.5 seconds slower (this includes a half second on the score screen from collecting red coins). Come to a complete stop just a bit to the right of the peak of the hill and upspit the rat. Then bounce on him to get an extended flutter to the ledge, skipping the stairs (see video).
The harder bottom route is shown in the IL record. If you have extra eggs from 3-5 you can shoot the first hedgehog out of your way instead of tonguing him. The bounce off the rat to skip pushing the rock requires you to press B just a bit after landing on the rat because if you bonk the ceiling you will lose your extended flutter. You can also spit a rat to the right of the final hedgehog and bounce off of it but this doesn't save as much time.
The mushroom platform that the key door is on is higher in the middle than on the sides, meaning that you won't be able to make it onto the platform if the peak of your flutter is too far right. Shooting the last hedgehog allows you to jump to the door from farther left, meaning you don't have to left-right to get up onto it.