2004 Lincoln LS Battery

My son replaced the battery one year ago but he did not remove the plastic cap on the NEGATIVE terminal. What problems can this cause? The mechanic said it could cause many different codes and make the car run funny as well with strange electrical problems but I have had no problems. Recently the car overheated and the mechanic said it was because the battery problem caused the electric fan to not run (however the fan is always on, and it's loud and has been for at least three years) and that it caused the thermostat housing to crack. I think another mechanic that replaced my exhaust system got his torch too close to the manifold when he was replacing the full exhaust. Anyway, it's costing me $800 to fix this but I don't have a clue as to what has caused the problem except for the mechanic telling me it was all caused due to a faulty battery connection.