Spectrum diagrams -- questions

For the simple waves, do we ever care about anything other than the frequencies and amplitudes?

Pretty much no.

One additional thing ... phase. ...

We can't hear phase differences between simple waves.

(But, in a slightly different way, the phase difference between the complex wave picked up by our left ear and the complex wave picked up by our right ear is one of the things we use to figure out which direction a sound is coming from.)