Can you build a house for your pet?

Unfortunately, you can’t. The game doesn’t give you an option to build a house for your pet or any real option to at least keep your pet WITHIN your manor’s fence. While you’re in your manor, your pet will be tracking you from outside the fence. The real dog house can only be purchased at the mall using fed credits (hence costing real money) and it's only available as a package with a husky.