I just completed the game, it's great!
Only feedback I would give is to add Z to undo a move as it's expected now in Sokoban-like games, and make the stick picking/dropping quicker (or instantaneous). There could be an animation, but I think the animation should be quick and not block the player's movement if they want to move right away. I often found myself feeling like I was stuck in place when trying to move after picking the stick, or moving twice as soon as the picking animation was completed if I had clicked again thinking it didn't register the first time.
Oh, and one other thing, I think it would be super cute if the dog started walking in the direction of movement when not carrying a stick, and turn towards the stick when it picks it up. It would have to be tested to make sure it's not confusing since the dog isn't turning when holding a stick, but I think it would help give it more personnality!
Other than that, I loved it! I can't wait to see where it goes in future updates!