I was curious if Rover sends an email to the client if they have not paid yet and time is running out on the booking
I always message the client if 24 hours has passed and they have not paid for the booking. I say something like, "Hi, Ms. X, I just wanted to double check that you sill want me to keep Fido on (dates of request). If so, you have X hours left to confirm and pay for the stay. If your plans have changed, please let me know as soon as possible."
When you have accepted a booking, the client has 72 hours to pay. Reminders are sent out. However, you may not be the only sitter the client has interviewed. You could try messaging them yourself to see if you can find the status. If the booking expires, it is easy to resurrect.