When lightning strikes a car, it is not simply charge separation that protects the occupants; the conductive skin provides a preferential path for a current. If the charge were to simply be deposited on the skin of the car, something exciting might happen, but in fact the charge is carried through the car to the ground. It is definitely possible to apply a large enough current that ordinary conduction no longer occurs, but this isn't because the metal skin runs out of electron/hole pairs -- it's because the skin heats up and vaporizes!
Your premise for part 2 isn't quite correct. If you stick an electron inside a neutral, conductive sphere, you can still see the charge outside the sphere. If the sphere is grounded, then you will not, but that's because you've essentially hidden the charge on the ground and given the sphere an opposing charge.
Conductive shells shield their contents from external fields, but they do not shield the external environment from the charges of their contents.
If you add charge, grounding will continue to cancel your fields. If you increase the charge rapidly, then you can gain a momentary field, but currents will eventually cancel the charge accumulation. (Essentially, you are charging a capacitor, which is leaky because of connection to the ground.)
To return to the question you posed in part 1; what happens if we apply a slowly increasing field to a hollow conductive sphere that is perfectly insulated from ground? At first, polarization will shield the interior of the sphere from the external electric field. The conduction electrons will flow to create a polarization across the sphere. Eventually, however, the material will breakdown.
Exactly how is a matter of conjecture, on my part. Because the sphere will polarize, with electrons accumulating on one side and bare ion lattice on the other, I hypothesize that the ends of the sphere will begin to smear out perpendicular to the electric field. The electric field balances the repulsion between like species along the electric field, but not across it. Therefore, the ends of the sphere will experience a shear that thins them and breaks them. These broken pieces, being charged, then fly along the electric field at colossal speeds, allowing electric field into the erstwhile interior, and badly damaging your lab.
The lightning rod is based on two principles theorized by Benjamin Franklin. Lightning dissipation theory, and lightning diversion theory.
Lightning Dissipation Theory
This theory says that if you point a pointy metal object toward a polarized cloud, the metal object will be able to bleed off some of the energy from the cloud. Thus preventing a lightning strike.
This theory can actually be demonstrated, using a Van de Graaff generator and a nail. This YouTube video demonstrates the theory.
While this theory holds up on the small scale, it's been shown not to be effective at dissipating the large amount of energy built up in a storm. Fortunately, the design of the dissipation device (lightning rod) is also a great diversion device.
Lightning Diversion Theory
The lightning diversion theory says that if you provide a preferable path for the energy to travel along, there's a high probability the energy will follow that path.
Lightning rods are designed to be the highest objects around. This puts them closer to the polarized cloud, and reduces the distance the lightning must travel through the air. They are also made from conductive materials, and are connected to the earth through highly conductive materials. This provides a low resistance path to ground, making it a preferable path for lightning to follow.
While both theories hold up in the laboratory, only diversion theory seems to offer a viable lightning protection system.
Best Answer
Yes, if it is not a plastic covered car it is an effective Faraday cage.
If the tires are such that the car is insulated electrically, if it is hit it will take some time to discharge to the ground, but still the passengers would be safer than standing next to it outside. I have learned that modern tires are particularly constructed so that the static charge generated by the friction on the road is discharged so that would also help. ( in olden times they used to have chains trailing from the trucks in order to discharge the static. Recently I saw a car with a discharger too, trailing on the road!).
Wrong, the current actually may start from the ground. That is the rational of the lightning rods, to create a path for a current to be generated by the potential difference to the cloud and to meet the current from the clouds in a prefered location instead of a random one. It is not wise to stand next to a rod, read in the link the amount of power dissipated by a bolt.
And it is not wise to stand, because you may also give rise to leaders that will meet the lightning path. If in the open it is best you fall on the ground as much sheltered as possible.
A colleague once was about 20 meters from a lightning bolt, and he was so shocked by the sound and fury, it took him a week to come down to normal.