Tell us about yourself!
I have not worked with RGB strips for this instructable but I my guess would be that each of those is a 12V line for R,G and B and one more for ground. If you connect all three to the same signal you should get a white(ish) light, overkill of an RGB strip for this project though.
I believe you are doing this by programming a microprocessor, and by looking at the code I guess it is an arduino. I remember you can do PWM on an arduino to get a fade effect so you could always do that. I think it will be all based on code. Another purely analog way of getting a fade effect is to put a capacitor in parallel between your pin and led strip.It gets charged when your pin goes high and once it goes low will let the light fade slowly as it gets discharged. You may have to play with capacitors of different capacity.
Infinite RGB LED Cube
The figure in step 10 shows how to connect the ends of your strip to the circuit, and what switch are you looking for ?
So you are trying to power your lights using the current from your audio device ? That's never going to happen and is probably going to fry your audio output circuit. I think you should consult some of the other instructables which are based on the TIP31C, the circuit is pretty straightforward for that.
The Arduino Internet Gizmo
MAKALU modular origami sculpture - 6 woven pentagons