If someone wants to book for an overnight tonight and an evening pick up tomorrow should they be charged for an overnight rate and the doggy daycare rate for tomorrow?

House Sitting consists of a 24 hour period according to Rover

Yes, that is the best way to charge it. It's not quite 2 nights, so one overnight plus a full daycare session should cover the difference.

It is entirely up to you! I charge per night, and don't charge daycare if they pick them up in the evening the following day. But I only accept one client at a time. I imagine accepting multiple clients may work differently.