iMovie IOS | Can I record video while simultaneously listening to an audio track in the iMovie project?

You can do that with the PhotoBooth app's internal camera on your Mac. (You said iOS in your heading, but since this is iMovie for Mac I will assume that you are working with the Mac.)

Start your audio playing in the iMovie for Mac project. Listen to it with earphones or an ear piece plugged into the mic socket. Then click on PhotoBooth, and click on video recording, and start singing as you play the audio. When you are done, stop the recording in PhotoBooth. Drag the completed video clip out of PhotoBooth onto your desktop. Then drag the video into iMovie and sync it up with your audio. The reason that I said to use headphones while listening to the iMovie audio is so that the PhotoBooth recording does not pick up the audio in the iMovie project as it is playing.

-- Rich