At what point do you reserve/block out dates requested?

I am "booked" and not available when a customer pays. Just because someone contacts you, there's no guarantee they are not interviewing several other sitters. Not all requests result in a booking. If these are first-time customers, I don't see any reason to hold yourself open. It is first come first serve. Proceed with the second request. If the first one gets back to you and you've committed to the second (with payment made), just inform them that you are no longer available for those dates and they should be able to find another sitter on Rover.

I am available until they click that confirmation button! If you think a client is about to have their stay poached from under them, I would send a last-ditch message reminding them to confirm their stay because another person is trying to book with you. Include a time limit, such as "please confirm the reservation by 8 tonight" to make sure there are no surprises or hurt feelings.