...take the damage insurance ? If I don't take the insurance from the rental co., does any damage to the car go on my personal car insurance even if it's not my fault?

Check with your companies travel dept. Our company is self insured so it is covered through them.

Yes.

If you are business, take the insurance.

If your insurance covers rentals, and you didn't get the insurance, it will go on your personal insurance regardless of the reason for renting the car. And yes, it is recorded by your company even if it's not your fault - and shouldn't effect your rates, but with some companies it does.