Almost true. The CD/Tape out are connected with your main outs, in other words it is a parallel output. That is why you can use either of them. They both catch up on the same "lines" going to your control room IF and only IF your CD/Tape Button is released. Pressed it will not go to the Control room outs (headphones/speakers).

I assume we talk about situation A here, since you mention two times the headphone in the mixer. So the situations are:

Checking input levels/sound (before recording) : release the CD/tape to ctrl button -> your input goes to CTRL out and you can set levels. (the LED is the most important indicator, your ears second indicator)

After setting the MIC input level for checking the sound being recorded together with the music(or mixdown situation): Pressed CD tape to ctrl button.

And yes that part you are right, with this setup you should solve the routing problem.

The compression/weird voice versus instrument problem is another thing which I explained as well how to check and solve.