@John6402 Firstly, with regards to reviews, you have been wrongly advised. You have 14 days after a guest leaves to write a review for them and they also have the 14 days to write a review for you. Both reviews will be published either once they have both been written, or in the case of only one party writing a review, once the 14 days have elapsed. It is important to write reviews for your guests, particularly if they have broken house rules etc in order to warn future hosts.
With regards to the cat, this is not a permitted/recognised service animal and you would be quite within your rights to ask the guest to either leave or remove the cat from your property due to breaking your house rules. I think you will really need to address this issue with regards to a pet fee sooner rather than later as I suspect you will find it difficult, now that you are aware of it, to try and claim fees in retrospect. However, you should be able to claim additional cleaning fees if necessary. It seems as though you have solved the other issues already with the guest (leaky fridge and WiFi) and hopefully you have all the communication with regards to this on the Airbnb Messages just in case the guest becomes difficult over them after check out.