Can I use Uber in airport mode?

You can, however only in a very special circumstance.

For example you got a passenger who got into your car, because they ordered you to come get them, then once they get into your car, they covertly Cancel your ride in hopes you wil lnot notice, but you do notice. You ask them if they cancelled their ride, and they say, no, must be by mistake. I fyou suspect they are full of crap and were trying to game the system you politely ask them to re-request (passenegr thinks they will get you again as the driver) sinc eyou are the closest driver. at this moment you secretly swith your phone to airport mode, because you do not want to be driving the sneakly bustard anywhere, and they get matched up with a different driver, so you kick them out:)

No, you cannot.  Not sure why you would want to.

Here's one way. You can turn on WiFi while on Airplane Mode. So do that, and connect to a nearby wifi spot. Then you can call Uber.

no, absolutley not.  what are you thinking.  You must not know anything about how your phone works.  I am surprised you even know "airport mode."