There are various scenarios but I think the one you are thinking of is you show up to the hotel, hand them your cash, and pay at the desk. Often times, though I have not been in a hotel that will not ask for a card, they ask you for a credit card and put an authorization on there in case you smoke in the room, or maybe turn it into spaghetti or any other random incidentals.
An authorization is basically the hotel telling the credit card that they're "reserving space" so to speak on your credit limit on the card.
Regarding your rental car experience, that's a common scam and again moreso seen in Europe, but not really anything to do with the method of payment. They would have just mailed you a bill for the damages instead. I guess you could ignore it.