I downloaded "d3dx9_39.dll", thanks to ericrevollo. It seemed to fix the graphics issue, but not when I used XAudio2. Hopefully someone here might happen to know the name of the file I'd need to fix that? I get the feeling that's the only thing stopping my game from running without actually installing the specific version of DirectX.
Including the Redistributable DirectX9 installer is most likely my last resort to solving this problem, though...