I have been to cozumel 5 times, and I am familiar with the taxis, location, prices etc...
My question is...Should we pack the car seat for the toddler? Do people usually do that? would that be "checked" luggage or "carry on"???

I think you will need to check with your airline. This is the policy for Delta/NWA.

Certainly a toddlers saftey is a personal thing. But it seems to me that my Taxi rides in cozumel were never very far and speeds did not get much over 30 mph if that on the city streets. It might depend on where you are going, but keeping a tight grip on your toddler might be enough for a short taxi ride.

Last time I walked the 3 miles back to the ship from a shopping maul of some kind, It was beautiful weather and I'm a runner anyway, it was lovely.