The answer is Yes, Yes, Yes. (I'm a rusty electrical engineer)
Anything else would be a violation of 2nd law of thermodynamics.
Let's start with a simple magnetic moving across a coil.
If you then put a load on the coil, such as a resistance, then swiping the magnetic will attempt to induce a current, which will be limited by resistance, and as it does so, the voltage will rise across the coil. (It takes time for charges to pass through a resistance, and a voltage is created as the charges accumulate) This is an inductive spike.
If you swipe a magnet so that an emf is generated in the coil and you have the ends of the coil unattached (open), then you will attempt to induce a current, but there will be no path for the current to flow, and a large separation of charge creates a much higher voltage spike from your coil.
If you swipe a magnet so that an emf is generated in the coil and you have the ends of the coil shorted together (!), then as much current as can be induced by the changing magnetic field is allowed to do so. Note that voltage will be LOW, current will be HIGH.
However, the high current induced in the coil ALSO produces a magnetic field, in the opposite direction as to the magnetic field change due to the swipe. In other words, if a magnetic is swiped, whatever current is allowed to flow in the coil will resist the swipe. This is how regenerative braking works. (Think stepper motors, or fancy electric cars)
In a car, this system works as a whole. The generator attempts to induce a current, but the current is resisted by the internal resistance of the battery, etc. In effect, an electrical system will only draw as much current as it needs (you can think of it as current draw is determined by the load)
Your car is "governed", meaning that it will use less gas or more gas in order to maintain its idle setting.
What that means is, if the load on the generator is greater, it will be able to induce a greater current to meet that need, at the expense of an opposite field by that current, meaning the engine will be loaded down and the gas usage will rise.
Now, you mentioned alternators ~~ these are a slightly different beast in that they don't have permanent magnets. Instead, the regulators in them shunt current through their field coils to create a magnetic field. In essence, though, the same result occurs.
The only effect that the regulator has (for both generators and alternators) is that there will be an upper limit on the amount of load that can be "seen" by the generator/alternator, in order to protect the windings from melting, and to maintain a more consistent system voltage.
An alternator which has no load on it will free spin, as the regulator shunts no current through the field windings.
A generator which has no load will --free spin-- [Edit: It won't free spin, because the permanent magnetic can still induce eddy currents in the core of the armature which resist the motion, etc.] , but generate high open circuit voltages. That gets handled with bleeder circuitry in the regulator.