When Bugs gets a carrot out of his safe deposit box, there are no keys involved - and the box only has one slot for a key. Usually two keys are required - one held by the bank, and one held by the renter of the box, for security.
When Bugs is pulled from the telephone booth, he is holding the earpiece to his ears, and telling the police where he is. They could not have heard him, because the mouthpiece was still in the booth.
The kitchen door in Rocky's hideout swings outward for most of the cartoon; it swings inward when the police break in.