(not in the aviation industry, just a fan, so grain of salt needs to be applied)

Aircraft age isn't in years, but in number of pressurisations (takeoff/landing cycles). A low cost airline optimising their schedule to squeeze every second of their airframe with constant short distance hopping around would result in much more of those compared to a legacy airline's widebody that is flown daily on a very long flight. So it depends on aircraft type/size and usage patterns, but Ryanair's 737 will be able to handle less years than DHL's 767Fs.

So for the 747 in the story, 22 years is on the long end, but not unheard of.

Do the climates a plane flies in and out of make a difference too, or is this such a minor factor compared to the act of pressurization that it doesn't matter? I remember flying out of Chicago once when I was a kid and having to wait for us to be coated in some special fluid because it was so cold that day that the takeoff was going to be especially stressful on the aircraft otherwise.

That fluid would have been deicer. https://en.wikipedia.org/wiki/Ground_deicing_of_aircraft

Climate can matter a bit, since it can exacerbate other issues and cause faster corrosion than expected.

It depends, but usually after a certain number of cycles, there will be a larger maintenance check (D-check, see https://en.m.wikipedia.org/wiki/Aircraft_maintenance_checks). This will include removing all paint and performing lots of testing for cracks. It's possible to extend the life of an aircraft as long as it is economically viable (by replacing parts and performing the necessary structural repairs).

Depends on the type.

For small non-pressurized aircraft that people do their initial training in, it's very normal to learn in a plane that is more than 60 years old.