In the first place, cooling systems are made to be full. Not having a tight (or in this situation – any) connection to the reservoir means every time the engine cools the radiator is sucking in air and when hot it will push out air or coolant.
You could have a headgasket problem but running the vehicle the way it is now will cause overheating and that alone could lead to a head gasket problem.
As for the alternator frying the battery, put a volt meter across the battery terminals while the engine is running. It should read in the 14 volt range or maybe less if at idle.
If the valve cover is leaking – new gasket.
Old Hondas are known for main relay problems. You have an old Honda.