How can I record audio & video synced with another audio file?
You are over thinking it.  Edit a quick beep in before the audio starts on the song, play the song aloud while you are recording the drums.  Hit one of your drums so that you can see and hear it quickly on the video as well.  Record the audio direct from your drums and have a third track that is the microphone for the camera.  The camera audio track will here both the drums and see the drum hit being used to sync.  Then align the song and the drum track based on those markers in the video editor of your choice.