Battery Light

load test the circuits at the alternator.have had alternators fail by turning on the battery light when they were charging just fine.the internal circuitry fails intermittently.Also for the heater core concern have the coolant checked for electrolysis.this concern may also be related to the charging system.if you have a weak or corroded ground cable,the electricity will find a path to least resistance,often times through your heater core! take a voltmeter ground the black lead and put the red lead in the coolant at the radiator,it should read less than 1 volt.