Does the alternator always have the same amount of mechanical drag on the motor, or does it vary with charging requirements of the battery?
Varies. Whenever it's an alternator, a hand-wound flashlight or the megawatts-sized generator they have in power plants, the amount of electric energy drawn from the output is taken from the input shaft (actually about 2-10% more, to cover the loses of conversion). When you disconnect every load from an alternator, the amount of mechanical energy drawn from the engine drops to nearly 0 (nearly, because some loses are constant, like those of the driving belt, bearings, etc).
If you go camping and plug a 12 volt cooler, a laptop for watching movies, 5 phones for the whole family, etc into the car's battery, is one consequence of this that the alternator will respond by dragging more on the engine.
Yes, exactly. Discarding the loses, when you connect 100W electric cooler, that's 100W less at the wheels. (If we assume that efficiency of the alternator is same as efficiency of the entire power train.)
Or, would it not have any effect on fuel economy?
It will have effect on fuel economy, but it can be easily hidden. Even load as large as 100W is not very noticeable when compared to even moderate engine that can pump out 40-80kW. 100W taken from the lighter socket that's roughly 100W less at the wheels. When you're driving, your speed will drop slightly. Now 2 things can happen: either you won't notice, so the total fuel economy won't be changed at the cost of slower driving. Or you (or cruise control) will notice, and will press the accelerator harder so the speed will be maintained at the cost of increased fuel consumption. And when you consider that your car AC is another mechanical load on the engine, that consumes power in range of several kW, few hundred watts can easily pass unnoticed.
It's a really simple math: as the energy conservation principle says: energy can't be either created nor destroyed. Both sides must be equal, so EnginePower = DrivingPower + AlternatorPower (+ AirConditioningPower).
Now, could an alternator that always draws same power even exist? Sure, it could. But that energy, if not taken away in form of electricity would have to go somewhere, which usually means wasted as heat. So our hypothetical device would heat up like hell without load and run considerably cooler when loaded to the max. A 1kW alternator doesn't heat up with the power of a 1kW space heater when idling, which shows that it doesn't waste that much power. And when the energy is neither transmitted away (eg. in form of electricity) nor wasted in the form of heat it can only mean that it was never there in the first place.