Does anyone else find the Rover interface for listing services clunky or misleading ?

Yeah, if you're trying to use Rover to do anything outside of the very basic dog sitting at your home or the client's home, it can get a little confusing, especially on the client's end. That said, you CAN get Rover to work for just about any pet-related service, which makes it a very flexible tool.

"When your dog stays with Pat"

I don't think this is misleading, since the pet will be with you, regardless of whose house you are in. It's simply a place for you to outline what your clients can expect for their pet's experience while they are gone. If you're concerned, your first line could reference that you are a travel sitter "While you're away, your pets will be cared for in the comfort of their own home..."

For example, my Price Per Night might be set to $35, and then Extra Dog be set to $5. Meanwhile the Cat parameter is set to $9, does that mean extra cats will be $9 each?

Since Rover is primarily a dog sitting site, the bulk of the rate structure is designed to accommodate dogs. All cats would then be "extra" cats. You could easily adjust the rate to provide a discount for multiple cats if you wanted; such as $9 for one, $7 each for 2, etc.

As for adding a complete rate structure to your profile, since you're a traveling sitter and encountering a different set of needs for each client, I'd hesitate to be so specific with your rates ahead of time. You may find yourself in a situation that requires a lot more effort than it's worth. If I were you, I would stick to what Rover provides as a template (first dog rate, additional dog rate, holiday rate...) and say that since some stays (and some pets) will require additional time and care, you will provide a custom quote based on a client's individual wants and needs after you've discussed their expectations of you. This allows you far more flexibility and control. However, it's up to you - you still have the option to offer a different rate than advertised after speaking with the client and determining their true needs.

I agree that the pricing structure for Rover is confusing to pet owners who want a visiting daytime pet sitter or dog walker at their home. A big missed opportunity for Rover and sitters who will sign up on competing services or go it alone. A simple solution is adding a drop down choice for a "per visit" price for dogs or cats. Easy solution for a repeated issue on these forums. I suggested it to Rover after my first week on the site.

"When your dog stays with Pat"

Yes, this means that the dog is staying at your place. "When Pat stays with your dog" means that you are staying at the client's home. It would not even mean a traveling sitter who visits a dog. I think your idea of a price chart posted within your profile is an excellent idea if you believe it would illustrate things better for customers. It probably wouldn't hurt to discuss this with Rover to see if there are any planned changes to the website or that your suggestion could potentially lead to one.