The stateroom hallways are wide enough that the
ECV should fit through if nothing has been left in the way. Stateroom hosts will move the housekeeping carts if necessary as they are always around; but if other guests have left
ECVs or strollers or food trays, sometimes though can create obstacles. You may have a wait for an elevator, though being on deck 2 that should work to your advantage. Be aware there can be a good sized lip to get through doorways to the outside decks.
Since you do not have an HA stateroom, it is unlikely that an ECV will fit through the doorway into the stateroom. So yes you may need to park in the "designated" parking areas. I think that is deck 2 on the classics or deck 6 on the Dream/Fantasy. If you cannot walk to/from your stateroom, you may need to have another member of your party act as "runner" to park and retrieve the ECV for you. Keep checking with
DCL as an HA room may become available for you.
I have not been to Cozumel or Jamaica, but accessibility in ports varies. Many of the countries on cruise itineraries do not have the strong accessibility laws that we are used to in the U.S. As such, there may not be curb cuts, may be steps into buildings/doorways, etc. You might try asking on the DCL forum about accessibility in those specific ports. You can take the ECV off the ship, be aware the ramp may be rather steep (I assume that depends on water levels).
You may not want to take the ECV off at
Castaway Cay. It's basically beach, and you won't be able to move through sand. There are manual sand wheelchairs if you have someone who can push. You can also ask for golf-cart assistance to get from the ship to the tram.
Out of Port Canaveral, I know that
Brevard Medical Equipment delivers ECVs to the ship, as does
Walker Mobility and
Special Needs at Sea. Most of the Orlando area rentals will not deliver to the ship, though depending on your pre- and post-cruise plans you may be able to arrange pick-up and return at an Orlando area hotel/resort.
Enjoy your cruise!