Why when a client books me to house sit, my per night fee isn’t applied & only the per pet fee is applied???

I've looked at your rate table and assume you've figured it out because it doesn't match what you've just described. In fact, it looks perfectly alright, without any anomalies.

First off, though, there is no base rate to house sit that does not include one dog, whether it is an adult dog or a puppy. If someone books you with a puppy, then you will be paid $65/night. If an adult dog, then $60/night. If they have 2 dogs, then they'll be charged an additional $38/night, so they'll be paying either $98/night or $103 if one of the 2 dogs is a puppy.

If you'll look at your rate table here, some rates have a plus (+) sign; some do not. That controls how the components are charged. The additional dog or cat is additive, on top of the fee for the initial animal. But the rest are not.