The movement mechanics are the same as Stephen's Sausage Roll. Due to the single-screen and meta nature, it would be hard to insert tutorial levels and unfortunately some compromisation has to be made . I would recommend you to play through the introductory levels in An Elastephantastic Adventure, which would definitely help you to understand how an elephant moves and push crates.
And for the checkpoint feature, room 3 not having checkpoint is to prevent you from pushing the crate to the dead end and getting stuck forever. Instead, you can use undo to backtrack the progress (even if you mistakenly press R). In other rooms, there are proper checkpoints.