My son and his girlfriend met me in Chicago on our way to New York a few weeks ago.... They put their pup (6 months old) in a soft-sided carrier. The soft-sided is good because you can mush it under the seat in front of you if a bit too tall... (try not to crush the dog!!
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I don't think the airlines really check the carrier all that much (for a certain brand/airline approved)... You carry the dog/pup thru the security station with you while the bag goes thru xray with your other carry-on luggage.
Have a nice trip!