There's a small booth just before you cross the bridge to leave Mexico. Hand in the FMM form that you filled out on the plane when you arrived. There are no departure fees; nothing to pay at the border. That's for visitors; I don't know about vehicles.
Visitors don't pay anything (assuming you're from the US or Canada) -- you should get a stamp allowing a stay of 30 days. I don't know about the vehicle, but you will have to purchase insurance to cover you for the time spent in Belize. There is a place (maybe 2) at the border, near the immigration building, where you can buy insurance. Ask the immigration officers.
Visitors/tourists pay BZ$37.50 (US$18.75) payable in BZ$ or US$ at the land border. Don't know about the vehicle.
Visitors must get a new FMM (visitor's form). Since you are in transit from Belize to another country, you must pay the MX$262 fee for the FMM. The fee is not paid at the border but at a bank before you depart Mexico -- keep the receipt with the form to hand in at Cancun airport. I've heard that Cancun airport is one of the airports where the FMM fee can be paid upon departure, but that's unconfirmed.
The spray station is a couple of km from the immigration building as you head towards Chetumal -- it's on your right.
Departing from Cancun Airport
I've never paid anything -- there is no departure fee. If there are any airport fees, they must have been included in the ticket price. If your FMM fee has not been paid, I don't know if you can pay it at the airport, or have to return to a bank to pay it.
With the car, an idea might be to find a hotel with parking in Corozal. Leave the car there while you go to CC. You could get the water taxi to Ambergris then CC, or you could fly from Corozal.
With a car, I'd say you should get to San Ignacio & Placencia. Have a drive through Hopkins.
I have the names of two Mexican car rental companies that allow vehicles into Belize.
Do you need them?