I used to fly with my dog occasionally, and I followed all the rules: 1) Required to bring a recent veterinary certificate of health 2) Pay $75-125 to bring the animal on. 3) Demonstrate the animal weighs less then 20 lbs. 4) Demonstrate the animal can fit in a soft sided crate that fits underneath an aircraft seat 5) give up having a personal item 6) keep the animal crated underneath the seat in front of you.

My dog never did his business on a flight (we never flew more than 6h and he was used to staying home way longer then that), but I had baggies ready to go.