I'm not sure about the humming sound...do you have a V6? It could be the "moosing" sound that the Duratec is prone to. Check out the FAQ section, they have a fix for it.
As far as your battery light coming on, it's probably your alternator. Under heavy loads, (meaning the pulley spinning fast) if it's not putting out enough power it'll drain the battery (Because the battery has to make up for the loss of voltage/amperage). It's most likely the brushes in it. If you have an Advance auto (or Autozone; I believe Autozone does this too), you can have your alternator tested.