The towing company must provide you with a receipt if you request it detailing the charges to date.
If your registration is expired, the towing company cannot require you to update it as a condition of returning your vehicle to them.
If you pay cash, the towing company should give you the cash change as accurately as possible. They cannot require you to pay the exact amount. If you are pay by debit or credit card, you should not be charged an additional fee for paying by card.