I won’t refer to your dog example, because the way it was presented is far too simplistic to depict photography as an art, not saying that the photo of a dog couldn’t be art.
A good photo as good painting, as art, cannot be defined or constrained by any set of rules, but it has to transmit something to the viewer, you may love or hate it, but that’s something intrinsic to any art work.
Art is all about breaking with the status quo and advancing by entering into path few people have dared to enter before. Most of times it means transgressing what “orthodox” photographers believe a technically sound photograph should be.
How do I know a photograph is art? When I see one because it tells me something different that most other pictures and it happens to be that many of them are done in a original way.
I know people like me that work both digital and film and we don’t have any respect for what “the authorities in the matter have stated as proper process” or any medium. We “laugh” at proper, because we know and handle it very well, but we want to break it to go beyond. The medium would be the canvas for a painter and the process would equate to the style, we, as them, are not constrained by them, they are mere tools.
I often step on negatives to get different random effects, make paper positives out of the negatives that I use to create paper negatives for contact printing (in the darkroom or in photoshop) and then scan to be bundled with digital raw photos in photoshop or the other way around. Making a artistic photograph goes beyond the camera, the medium and the process, everything is valid if it has a purpose.
Check the British Journal of Photography to see what people that have been working on the edge for more than half a century have been doing and where art in photography is going. This photography magazine probably the only one worth subscribing nowadays.