Looking at the door scene,
it's pretty clear that Lotso pushes it open. We can all push open doors quickly, but that doesn't make any of us 10-A. My late cat could push open unlatched doors with his head. I've also seen dogs (Both big and small) push open doors either with their heads or by standing on their hind legs and pushing with their front . Given the context of the film, only the doors to the classrooms are locked, not the ones that lead to the bathroom.
Those doors also don't appear to have a proper lockset, as Big Baby opened it all by himself, and he isn't an acrobatic type like Buzz or Jessie.
Watching the clip, I don't see the trash lid vibrating when Lotso slams his cane, and breaking a small plastic heart isn't really that impressive as any of us can do the exact same thing with zero effort.