Need to address one problem at a time. 1st, if you let it sit all winter without being charged or on a tender, you may be kissing that battery good by. But get a tender now and hook it up and let it sit until the green light comes on. Need a tender anyway..get get he cheap one. That may take two days. If after three days it hasn't fully charged... better consider a new battery as that one is too sulfated.
On the overheating, sounds like the coolant level in the radiator is low enough to expose the fan switch so..check it...not the recovery bottle, the radiator. While there look down inside and see if there is any sign of jelled coolant. This was a problem with their older coolants. This would also explain your cooling issue.