Okay, in 2015, we moved 800 miles away, and we had to fly down 4 pets. We bought 2 carriers, and did it in stages. Now, we flew American, so I can only attest to what they did/required of us, but here goes:
(1) We paid $125 per pet for the privilege. In addition, you give up your carry-on by bringing on your pet.
(2) The pet must be in an FAA-approved, soft-sided carrier that fits under the seat in front of you. The pet must weigh less than 25 pounds.
(3) Pets must come out of the carrier to go through security. I had to carry our cat through the scanner. As an aside, I stuck behind a guy who (naturally) had to try about 6 times before he got pulled out of line. All while holding a large, scared cat in a crowded airport.
(4) For one leg of our trip, we were on a puddle-jumper flight. We were still required (and able) to put our pets under the seat.
(5) We had to provide some vet paperwork (vaccine records, maybe?) before boarding
(6) Only so many pets are allowed onboard each flight, so we had to book way in advance. You get special boarding papers that you have to present when you check in.
I want to emphasize--all of the above was fully spelled out on American's website. Things might have changed a bit in 3 years, but I don't see them getting less strict about pet travel. You can't just decide today that you want to fly your pet to Aunt Hilda's for a visit tomorrow.
I don't believe for a second that the FA didn't know they had a dog, or thought she was doing the right thing. She may well have been trying to do the expedient thing to get the plane onto the runway, and may not have realized that the dog could die. She should have taken a few extra seconds to find another bag that could go into the overhead bin, to make room for the dog under the seat. And even that surprises me--IME, it's the overhead bins that get filled up quickly, not so much the floor.