Hi there! I fly with my baby about every few weeks. You definitely need his health certificate from your vet and proof of his rabies vaccination- they do not always ask to see it but if they do ask and you don't have it you won't be able to travel.
-The
nutrical they generally allow you to bring with you even if it is over the 3 oz limit [they have always let me anyway], I always bring
nutrical with me because sometimes they refuse to eat for a while during and after a flight.
-The purse that you mentioned I do not believe they would allow, and it wouldn't be a good option even if they did allow it. The carrier goes under the seat in front of you on the plane so I would think it would be very uncomfortable to sit for hours in a purse and likely wouldn't be very balanced and would tip over during the flight.
The carrier needs to be able to have room for your baby to stand up and turn around and lay down comfortably, it also must have good & proper ventilation, the carrier bottom must have some absorbent material or you can lay a blanket or towel on the bottom. If you are planning on your baby flying with you in the cabin the carrier must be able to fit underneath of the seat in front of you.
It is true that they do not allow your pet to be out of the carrier, when none of the flight attendants are around I always unzip part of the carrier and just check on my baby or give him some food and water.
Going through a security checkpoint you are not allowed to have any open bottles of liquid over 3 oz such as a bottle of water but what you can do is just have an empty water bottle and fill it up after you get through security.
The leash and harness isn't really necessary but it does come in hand for longer flights or when you have several flights to catch that way you can put the leash on and take your baby for a short walk in one of the designated areas at the airport for pets, or if the airport you are at does not have a designated pet area then go into a handicap stall in a restroom and let him/her walk around and stretch [this is also when the disposable pee pads will come in hand.
The first flight with your fur-baby will have you very nervous but try and stay as calm as possible because when you seem nervous it will upset your baby and he/she will also become nervous and stressed.
When your flight is over your baby may seem a bit lethargic and take a day or maybe two to return to his/her normal self- some animals are very resilient and take no time at all while others just need a little break. This is completely normal, but try to be sure to give an adequate amount of food and water. If your baby doesn't eat or drink for any period longer than 12 hours contact a veterinarian immediatly as this is when dehydration sets in. Good luck & have fun!