Yes, you are correct. Copyright does not have to be applied for. It belongs to the photographer, unless he is working under contract and it belongs to his employers. If two or more people are involved in taking the photo, the copyright belongs to all of them. In the case of the monkey, the owner of the photo has no claim to copyright as he did not make any intellectual contribution to the taking of the photo or its composition. The argument that he provided the equipment does not make any difference to ownership of copyright, legally and also logically - you might as well as argue that the photo could not have been taken without the manufacturer of the equipment - the lens, the sensor etc.