On most airlines, a health certificate is only required when a pet flies in cargo. If a health certificate is required, it must be dated within 10 days of the flight - so you cannot reuse a health certificate unless you plan to forge the date.
As far a head popping out goes, it isn't up the airlines, but rather it is an FDA rule that pets carried on board be in a completely closed carrier and kept under the seat during the flight. Most flight attendants do not have a problem if you put the carrier on your lap (after take-off and then put back to landing) provided the carrier is contained on your lap and not encroaching upon a passenger sitting next to you. That being said, most flight attendants will insist the bag stays completely closed. The only exception to this is if you are traveling with an emotional support animal or a true service dog.
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