If the alternator isn't charging the battery, did you have the alternator tested before you brought it home?
If the Battery light is going out, after start and the light stays out, what indications are you having?
Have you measured the Voltage on the battery?
Now what with the Grounds and see my signature....
Does your Battery light i the dash ever come on, at key in, start, then goose the gas a wee bit over 1100 rpm...
If the Alternator light goes out after you increase the rpm above 1600, but comes on when it falls below, 1600, then you have a rectifier issue.



