I'd start with Stanton wool. The pad I go with is called Smart Sense, and if needs to be seemed, I have the installer duck tape it. Accidents do happen and should the rug get wet the pad will not. That's important because recurring stains come from the pad. You clean and clean the stain and it keeps coming back. Goal is to keep pad dry.

I have many Stanton carpets that I have used around my house. I have been very happy with the wear, no perceptible shedding, ease of cleaning the wool, and feel under foot. Definitely second oncapes suggestion.

I prefer a 3/8" rubber waffle waterproof pad to protect the carpet backing from abrasive wear and the subfloor from moisture. Enough for tack strips. That way you can use a carpet shampoo machine as much as necessary. Forget pad as a way to add softness. A foam pad quickly gets flattened to no benefit. All that comes down to carpet density. Face weight in ounces per square yard and pile height in inches from the backing. Face weight x 36 divided by the pile height. 4000 + will give you the softness underfoot. Nylon will spring back. Polyester will crush. If you can afford wool in your bedrooms that will beat nylon. For 55oz face weight the pile height is 1/2". Shaw Tuftex 'Enchanting' Anso nylon is one. Stainmaster Pet Protect 'Piedmont' or 'Paws Down' in 0.5" and 55-58 face weight is another. Ask for the Architectural Spec sheet to verify face weight, fiber makeup of Nylon 6.6 and pile height. Going to 0.65 pile height will require a face weight of 72 oz.