If I up my nightly rate, do my pending clients get charged my old rate or my new rate?

Anything that is already booked will remain at the price agreed upon by you and the client at the time of booking - those transactions have already completed, so the price will not change. Any pending (requested, but not yet booked) requests will also reflect the rate that was in effect when the request was generated unless you manually change it. Any new requests you receive after your rate change will reflect the new, higher rate automatically.

I believe the rate in effect the day they first contact you with their request will hold. If they cancel and try to re-book, the higher rate will be charged. I also have raised my rates lately and none of my clients have been affected.

You can always edit your rates before they are booked. But once they are booked, you would have to contact Rover customer service.