Pauki, Your question seems confused and the replies aren't clearing confusion.
Cozumel is an island, you seem to have gotten that part. Isla Mujeres is an island very close to Cancun, could you be confusing those two?
There is no ferry service between Cancun and Cozumel. The car ferry connects Cozumel to Calica which is south of Playa de Carmen. Playa de Carmen is located about one hour south of Cancun. No you can't commute between Cozumel and Cancun, that I would rate as nonsense. Figure a commute time of two hours minimally and the cost would be about $200 pesos each way. Only with established residency (figure some months) would the passenger ferry have discounted rates and one way cost to/from Cancun would drop to about $125 pesos.
I question if you are aware of the realities of life in Cozumel this time of year. The least of your concerns might be that hurricane season is about to begin. The months of September, October and then November are the absolute dead season economically. Hotels have mass layoffs of staff and many resorts dip below 20% occupancy rates. Retail stores struggle for revenue to be sufficient to pay their electric bills, paying rent is often a fantasy. People that depend upon commissions, they just don't get many. Taxi drivers sit and wait for hours between fares.
Will no one comment on the plan to bring a Great Dane to the island? Do you have a place to live and do you have a real salaried job? Cozumel can be a great place to live if you are retired, own a home and have a substantial retirement income or live off investments. It is a real tough place to make a living.
No Mayair would not be an option at about $1,000 pesos round trip and limited daily flights, It would be another 30-50 minutes to bus from airport to the bus station in Cancun