Also, the current consumption will not increase when the voltage drops. I=V/R, the resistance of the load stays the same, as the voltage drops, the current will also decrease.
Current consumption does increase as voltage drop. You are considering the wrong part of the system when you are applying equations. The amplifier tries to keep its power output constant, regardless of the voltage it is getting from the car. For it to maintain a constant power with a dropping voltage, current must increase. The real problem that is caused by this is that as current increases, the voltage drop from the resistance of the power delivery path increases which increases current requirements further and on and on and on. If the power supply transisitors and diodes could handle unlimited current, voltage wouldn't matter as long as you could get enough current, but since these devices can only flow a finite amount of current before they exceed their power dissapation rating, most amps are equiped with undervoltage protection circuits that shut the amp off before this can occur.
And lastly, power is what causes heat, P=V*I.
True, but probably not in the sense that you are thinking. Power is lost as heat. As current flows through a resistance there is a reduction in voltage across that resistance expressed as V=I*R. When you are figuring power lost as heat, the voltage drop is the voltage that is used for the calculation. So if for instance, you are getting a 1V drop over a run of cable at 100A you are losing 100W over that run of cable. Cable is not very good at shedding heat and that heat will build up and start to cause problems. Firstly resistance increases with heat. More resistance equals more loss and more heat build up. Basically, having an insufficient power delivery setup is a self feeding downward spiral. Not only will it cause poor performance from your audio setup and serious dimming problems which are annoying, but if it is severely deficient, it can cause damage to components and/or the car.
A few more points of clarification. The alternator is designed to be the primary source of power for all things electrical in your car while the engine is running. They are speced to the stock electrical needs of the car as it was delivered. This headroom wasn't meant to allow you to add a stereo. It was meant to allow the alternator to operate well within its capacity while supplying power for all the accessories built into the car and still be able to recharge the battery from the discharge required to start the car. That is the design goal behind it all.
When you add a high power stereo to the equation and don't do anything to upgrade the electrical system (even something as simple as upgrading the big 3 to minimize wasted power) you have upset the apple cart. At low-med volume and with a light accessory load, there is not a problem. The lack of an accessory load gives the stereo some power to work with without taxing the alt. At this point the battery is just another load (though a very small one) on the alt and the alt is suppying all the power needed by the car and stereo. Now switch over to a hot summer night. You've got the AC up the headlights on and the stereo going. This is the highest accessory load that the alt was speced for and now you have added the stereo draw on top of that. The result is now that when the bass hits and the stereo is drawing maximum power, the alt can't supply that power at the normal voltage. As a result the voltage drops (constant power) with the increase in current demand. Once the voltage drops to the the discharge voltage of the battery, the battery begins to contribute its stored power to the equation and staves off a further instantaneous voltage drop. The battery can easily supply several hundred amps of current for a short period so this is good. The problem is that as soon as the demand goes back down, the battery is again a load on the alt and since it is now partially discharged, it is a larger load than before and stays a larger load until it is fully recharged. If this was a single musical transient, then, no problem, but if the situation keeps repeating itself without enough of a lull in demand for the battery to recharge between each spike in demand, you will eventually get to the point where the alt cannot supply power for the car, the stereo and the recharging of the battery all at once. At this point the battery is in a state of perpetual discharge until the load is reduced to a point that the alt can again fully handle.
Getting to this state can have several negative consequences:
1) During this entire time, the alt is working at its maximum capacity. It isn't designed to do this. It will get very hot and it will shorten the life of the alternator.
2) If the battery isn't a deep cycle, discharging it in this way will shorten it's life greatly.
3) Continually running your amps on a lower voltage, while it may not hurt them directly, isn't helping their life either. They're going to run hotter and stay hotter.
In this vein, the big 3 can help with dimming problems. By reducing the voltage losses between the alt and the battery (the basic distribution hub for the car's electrical system) operating voltage will remain more constant during high curent demand sutuations. Also since less power is pissed away as heat, more is available for other uses.