1&2> You can claim it either from Credit card or your own insurance. - @ the time of insurance renewal they have this qs, asking whether you had any claim in past 3 or 5 years.
3> No clue
4> There is an option to get Grand Touring Advantage in your car insurance, that costs less than 5$ a month. Whatever coverage you have in your car that automatically gets extended to rental car.

Somebody working in rental company told me : When you are renting car, 3rd party is always covered irrespective of if you take their 20$ insurance or not.
The 20$ insurance will cover the rental car only.