Is it possible to edit different voices recorded by one microphone in Adobe Audition?

Crossed over with Steve:

Yes and no. It will depend on the distances involved. Also the further away from the mic, the more different the tone of the voice and or echoes or room tone.

This may help: https://www.youtube.com/watch?v=zXNxmNgt4CM

I would stress that at least get the people as close to the mic as possible.

Please try a test before setting out on a complicated recording.

In Effects, under Amplitude and Compression, there's an effect called Speech Volume Leveler - but I have to warn you that with singing, it might not be very effective, or sound particularly good. But it does do what it says on the tin though - it levels all of the volume in a file, pretty much destroying all of the dynamics.

For actually treating individual voices, you want something like Melodyne. But for this sort of purpose, it's a steep learning curve.