My question is about the pay rate? I want to know if we can charge night and day?

You absolutely can charge for the overage. I would and do all the time. Since I don't offer daycare, the amount I charge depends on the overage and I'll add it onto the original bill as an adjustment. If a dog is here 10 hours past, then I will charge another day. That is why it is important to get the dropoff and pickup times before the stay is finalized. However, you always have the "extend a stay" function to bill after the stay is already in progress.

Boarding at a kennel is like a hotel. Home boarding I think is a little different. I don't have established check-in/out times, but I won't accept any dogs before 8 am but will allow pickups up until 10 pm because I'm more a night person.

You are an independent contractor and can run your business however you want. Rover's FAQ state that each "night" of boarding is for up to 24 hours of care. If you want to charge early/late dropoff fees or a daycare fee for the time over 24 hours, then go right ahead. Always make sure your customers understand before the stay begins and try to get the additional charges onto the initial bill. It's just easier that way.