Your project in very intriguing and unusual and I have to commend you on your efforts so far. Well done!!!
As for being able to stop the music when cranking, the best method would be to pause the CD rather than mute it. Reason being, is that the CD will continue on regardless, therefore when cranking it up again would make it sound as if it skipped some of the music. If you're satisfied with that arrangement, then muting the player would be the way to go.
If you'd rather pause the CD, then you would need to make sure there is a pause button on your player. If so, then you would need to open up the facia of the player, locate the tactile switch for the pause button, analyze the way it works by studying the switch's tracks to see if they go to ground when switched. You'll find that 9 times out of 10, it will get grounded. You only need to attach one wire to the high side of the switch and that will then work as a pseudo mute.
As for the 6 second pause when re-cranking the lever before the music comes on, the picture below maybe the better way to go.
The main problem in using a switch to get the music going, as you've stated already, is the fact that you may stop the crank right on the point of it being switched. I thought you might be able to utilize a hall effect sensor arrangement using magnets, but the same may occur with that also. Whereby the magnet may stop and face the sensor. Same would happen with using an opto sensor arrangement as well.
Cranking a DC motor seems to be the best option, but I cannot tell you what delay in re-cranking would be, but I'm certain it would be much better than the switch arrangement.
When cranking the motor, it will produce a voltage that is limited by the 5.1V zener diode, as to not overload the transistors base if the crank is being turned at a reasonable pace. The transistor will be turned full on by the 5 volts at its base, thus switching on, thus energizing the relay and grounding the MUTE or PAUSE.
Any hobby DC motor would more than likely work well. Or if you can get one similar to the one in the link would be ideal:Jaycar
I hope the above helps.
Please keep us all posted on your progress.