I just got my Shadows of the Empire CD, and the CD audio plays without issue in DREAMM. So it looks like the WinMM wrapper is not properly supported. And from there, I found other issues in X-Wing and TIE Fighter.
In TIE Fighter for Windows, selecting the Joystick Flight Options doesn't show any options at all; the list is blank.
In X-Wing for Windows, the joystick buttons can be reassigned without issue. However, rotating the Z-axis causes changing the engine speed, when this shouldn't be doing anything, and indeed does nothing in the joystick configuration.
I can say for certain that these are issues with DirectInput emulation. It's serious in this case, since these games won't run at all without a joystick.
EDIT: Indeed this is an emulation issue, as playing these games directly from Windows doesn't have such problems.
The DOS CD version of X-Wing has a tendency to hang when entering a mission.
EDIT: The DOS versions of X-Wing and TIE Fighter work just fine with the joystick now. It must have been a configuration goof on Windows' part that fixed itself.
The joystick I'm using is the Logitech Extreme 3D Pro, if that matters. It works wonderfully with the Rebel Assault games.