To my knowledge, the Flex SDK doesn't require AIR.
Vast swaths of Flash-related tools require Java, though.
Flex includes mxmlc, which is the AS3 compiler.
FlashDevelop is an entire IDE and probably also includes mxmlc.
I come from the Realm of the Mad God private server community. It is a 2D game with some 3D graphics.
To my knowledge, instead of FlashDevelop, the people here use an old version of IntelliJ with flash support + a chinese activation server to edit the AS3 client code. IntelliJ's "build this solution for me now" button simply configures Flex SDK and shells out to it.
So the least faggy way to go about it is to use your favorite text editor or IDE to code it, then build with a shell/batch script that runs Flex with your preferred configuration.
Good luck finding useful AS3 and Stage3D documentation, examples and libraries nowadays, though.
Most recent private servers use the AIR SDK to build an exe+swf to use native extensions (such as Discord RPC) and to allow Polish kids who don't know how to use the projector to play. Savvy admins will change the .swf's extension to .dll or .bin or something so that the Hungarian kids aren't tempted to run it in the projector or jpex themselves a godmode and killaura.
HARMAN (Samsung) maintains AIR nowadays, but they added a splash screen watermark to the output after they took over. It can be removed by paying for a license or by hex editing the builds. It's probably not discontinued for good yet.
A guide to rawdogging mxmlc:
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