If a Dog owner dont pick up his dog for several days, do the app charge them automatically for those days?

No, it does not. If a dog isn't picked up and you receive no communication from the owner, then you need to contact Rover Customer Support at [Edit: Rover’s contact options have changed. Visit the Rover Help Center at https://support.rover.com/ to find the phone number, help articles, or chat with the team].

If you are in contact with the owner, you can easily send them a a booking request. But you will need to contact Support if you've waited and the days have passed.