How long do rechargeable handheld fans last on a charge?

Most fans last 4-20 hours depending on speed setting and battery size. Expect 8-12 hours at medium speed from a 4000mAh battery.

Are handheld fans allowed on airplanes?

Yes, lithium-ion battery fans under 100Wh are allowed in carry-on luggage. Most handheld fans are well under this limit.