Request a certification from your vet that your dog has been vaccinated for the current period for rabies and distemper. At the airport in the DR you must ask for the animal control person (Sanidad Animal) in the customs area. They will examine the paper the vet gave you and fill out an official entry form or permission, for which you must pay RD$100 (about US$6). Note you will have to make a special reservation for the cat to board with you and the travel cage will count as carry-on luggage and should fit under the seat. The airline will most likely charge you a fee for carrying the cat.