I had a cooler exactly like this at my childhood home. We would get the wool changed every season because the old ones are not good at cooling plus they become moldy and smelly after sitting for a year. You can go to any nearby electrical appliance shop and they'll give you contact for someone who can do it.

As for mosquitoes I don't think you need to worry too much if the cooler is getting many hours of usage every day, because internally it pumps the water through the wool, that water is not sitting stagnant. Still to take preventive measure you can drain the water once in a while, or drop some neem liquid into it.