Surely not? Cars are routinely driven by people who are not owners, and liability for traffic offences (including that the vehicle must be insured) is with the driver.
In my experience typically only minor infractions like parking violations are assigned to the registered owner of the vehicle, but in other case – accidents, running red lights etc. – the driver is liable regardless of who owns the car.
No. The general rule is that negligence is required to be held responsible. If I let my next door neighbor borrow my car to go to the grocery store, and he hits someone, I'm not responsible. Unless, the person can prove "negligent entrustment", i.e. it was irresponsible just to let this person borrow my car, e.g. they're a habitual drunk, or blind, or 11.
However, most auto liability insurance covers whoever you permit to drive the vehicle, so the owners policy does typically cover the fender bender on the way to the grocery store.
Correct, the owner's insurance policy is the primary coverage when the owner lends their car to a 3rd party. Obviously in the case of a moving violation the driver is at fault and receives the penalty, but damage is still covered by the owner's policy. In the case where the other driver is at fault, that car's owner's insurance is liable.
