Can a credit card company process a charge against someone for a car rental accident if it puts it 1000's over the credit limit on the card?

They can try, but your credit card issuer determines how much over the limit you can go based on your credit. For most people, it's not much.

Generally speaking, you will be billed for the amount and it will go to collections and judgment if you don't pay. If you owe, work out a payment agreement. And I agree with the other poster - most credit cards offer insurance for rentals.

Yes, it can, but don't most credit cards offer free rental car insurance?

No, but they will just ruin the person's credit and they'll never be able to rent a car or apartment and never be able to get a loan or mortgage ever again.

No the card should not accept a charge that much over limit. This doesn't mean you won't owe the money.