Whirring sound and battery light on

If it's a rapid machine gun like clicking it is probably the starter motor dying. Wiggling the wire suggests an electrical earth issue. Disconnect the battery. Disconnect the starter motor wire at the engine end and rub the two faces that touch with fine sandpaper. Refit the wire and the battery.
You should also get the alternator and battery tested as the battery light and heater issues could be due to a charging or battery problem.

Power steering reservour may have plugged screen. Fluid flows better when warmer weather thus, no noise of starving power steering. Have a look at this link. http://www.youtube.com/watch?v=_QjBLYltLYY

Does it change with vehicle speed or engine rpm?

vehicle speed: check wheel bearings lift vehicle and support it with jack stands grab the wheel at the 12 and 6 o'clock positions( to rule out steering linkage play) and wiggle in and out. A bad bearing that loud will have some play.

remove the belt and see how much play is on the alt pulley sounds like a bearing wearing out.

Have the power steering pump checked for excessive noise by a shop locally.

check fluid levels specialy the power streering if its alternator your charge light should be on also check waterpump make sure there is no play in water pump pully or any leaks

COULD BE A NUMBER OF THINGS OPEN HOOD TRY TO PIN POINT WHERE ITS COMEING FROM

If the check engine light is on, take the vehicle to an autoparts store and have them use a scan tool to retrieve any fault code(s) stored in the computer. Most parts stores offer this service at no charge. The fault code(s) will help narrow down the problem/potential problem. Sure hope this helps and best wishes.

Sounds to me like a dry, loose or worn serpentine belt. With the motor idling, lightly spray the serpentine belt as it's spinning with WD-40 or other oil based lubricant (Don't spray near the alternator). If this doesn't work, turn the mortor off and check the tension and for cracks or worn spots. The brakes will not squeal when you are completely still.

It is your power steering pump. The solution is to replace the P/S fluid resevoir and fill or flush the system.