Adding onto this before I forget:
Don't read further due to spoilers:
This is how I solved the game with 22 moves, and 30 moves for 3 trophies.
For the first few levels, you can cheese a little knowing codes in advance, or by carrying/throwing ladder to pass doors.
For level "Grotto" you can clear it by throwing both the red key over the door, and throwing the ladder over to the door. Then, carry the red key into the next level, this is a "key" part to solving frame 9, opening the red box for getting the green key without needing the red key on this level.
Snapshot 14 and 15 are a bit of clever use to get a ladder, green, and blue key to the top of the level, the ladder is used to solve the next level without needing time travel.
Snapshot 14 is also used
The rest of the solutions are relatively normal and just using the ladder-climb-over-door trick.