Well to be technically correct, an engine doesn't "suck" air in. As the piston goes down it tries to draw air in. The throttle valve creates a restriction therefore causing a vacuum or negative air pressure state. The outside air, being of higher pressure 14.7:1 (at sea level) tries to force it's way to the lower air pressure area. It's actually atmostpheric pressure trying to push it's way in to the engine... the engine doesn't "suck" in air.

You're right, it probably doesn't draw more power than your stereo system... you have no noticable power loss.. well, with an 'electric turbo' you'll have no noticble power gain.

As for the cold air intake idea... that is a totally separate issue. It's a known fact that hot air is less dense than cold air. Draw hot air from under the hood and you don't have as many air molecules entering the intake as you do if you draw fresh "cooler" air from outside the engine compartment. More air molecules allows the fuel to burn more efficiently which will create more power.

With the electric turbo, unless you can compress the air going in, you won't get any more air going through it than what atmostpheric pressure can force in. The electric motors that small just aren't capable of creating boost in the levels high enough to make any noticable power increases. Also, keep in mind that when you compress air it gets hot. Hot air has less molecules than cold air... that's why turbo chargers 'usually' have intercoolers on the intake air stream after the turbo.
They use the intercooler to help cool the compressed air.

I'm not going to sit an argue over something that has been proven time and time again that it doesn't work. If you wish to go waste your money, that's fine with me. It's your vehicle and it's your money. If it was so inexpensive to do and it worked, then why don't the manufacturers use them instead of using exhuast driven turbos or belt driven superchargers? It sure would be cheaper, don'tcha think?
If they really worked, I would think they would be sold in performance and speed shops, rather than by someone on Ebay..

Oh, and as far as running straight from the battery to the switch, well, last time I checked, the alternator charges the battery. The heavier the load you draw on the battery, the alternator has to work that much harder to keep the battery in a charged state. It wont' matter where you pull the current from, the alternator is the work horse that has to replace the current.