How big a dog?

Shade would be helpful I should think. I like Mesquite trees.

I like hardscape for the civilized desert look so raised beds work to add interest, make gardening easier and generally dogs stay out of them. I like Pergola type deals too.

A pondless water feature is very nice imo for a civilized desert look and the dog would always be able to have a drink. Some like to lay in them to cool off.

For the ground I like decomposed granite. Dogs don't seem to mind walking on it and they don't seem interested in digging in it, but that depends on the dog. My dog likes to dig in soil with compost for the smell I think. Manure is like cat nip to them. No interest in dry clay covered with decomposed granite. Their feet don't get too muddy in it.

Not really a desert look but fake grass as come a long way. My neighbor just installed some for her boxers.

A cat friendly backyard is one a cat never goes into imo. Just because cats roam widely and generally it's just a question of time before a car, dog, other cat, or whatever gets them. But if you really want the cat outside I can tell you they love water features that attract birds. Cats love to kill stuff so that's always fun. Some cats like fish ponds, but others seem to show no interest.

If you want more of a lush civilized desert look you can do bog beds which are basically small ponds filled with soil. Plants that like wet feet can then be planted. Very low water use and you get impressive plants and flowers with very little effort. Plants stay contained. In AZ I've tried Canna and sedges and both worked very well. There are others too.