It depends on the airline...The safest thing to do is to contact the airline, I'm sure the info would be listed on there website too. They all have different rules. I'm not sure if the juicy bag is approved. I know that some airlines sell carriers...not sure about the quality though, and I'm sure Izzy would be more comfy in her own bag that she is use to. You should really call the airline and book Izzy on the flight, on Jetblue I think it was $50 for Peanut. They have limits on how many pets can ride in the cabin so call soon, I doubt you want Izzy to end up in cargo. Good luck on Izzy's first flight!
With the bags, I think most airlines are fine with the carrier as long as it fits under the seat, and it is able to be fully zipped. Jetblue also checked to see if Peanut had enough room to stretch out, stand, and turn around.
I forgot to say that most airlines do need to have a letter from the vet within a certain time period before the flight.