Cats

If you have the money, I would recommend a Litter-Robot. They also sell refurbished/older models for cheaper. I have the LR3 but I think now they are on 4. It is amazing! Haven't had to scoop since we got it. It serves all 4 cats in my home just fine. For me, it was TOTALLY worth the money.

Before that though, I used those big plastic storage bins with the lid. I would cut out a big hole in the lid for my cats to enter/exit. This cut back on litter getting tracked around the house like with an open box/tray type box. Plus it offers a little more privacy.

For litter I use Dr. Elsey's! Good luck to ya!! Cats are beautiful, funny, fantastic creatures.

If your cat will use it, and you have the oof, a Litter Robot is very, very nice. It addresses every one of your requirements.

I like covered litter boxes with the door flap. If you've never had cats before, one thing to keep in mind is the number of cat boxes that you'll need. Typically people recommend N+1. For two cats, I would have at least 2 litter boxes. Besides, cats like having options. With respect to the kind of litter, avoid lavender-scented litters as lavender is a natural cat repellant. Also avoid silicone-based litters as it dries out their coats.

You could also try a top-entry litter box to keep the dog out of it. I've had success with using baby gates to bar the dog from the catbox area.