Power steering not working

Have you checked the level of your power steering fluid?

Do you have electic power steering?

Check the power steering fluid levels. Also the pump may need replacing if it isn't delivering enough hydraulic pressure to the steering box. Be sure to lube all the steering components on the chassis if possible.

for the whirring noice, check for the power steering fluid level, if it's low refill with power steering fluid to the propper level. If the level its fine and the noise continues you have to tighten the engine belts via the tensioner bearings. As for the battery light it's probably letting you know that the alternator is failing, when the light comes on it means that in that precisse moment the alternator isn't charging the battery and that could lead you to a dead battery. (The alternator has a line that it's in charge of activating the battery light if this cable is loose that could make the light go on, secure that cable and if the light never comes on then the alternator is fine) Hope that helps

If you think the tyres are in reasonable condition get the wheels balanced. If the front tyre/wheels are out of balance you get shake in the steering wheel at certain speeds. Backs out of balance are usually felt through the seat.
I cant see a connection between the power steer fluid level and the gear shift.

that wold be a loos belt. you mit wont to check for cracks if the belt is old or worn you mite wont to replace it.

Make sure that your vehicle is parked on a level surface and the engine is cool before you check fluid levels. Power steering fluid and other fluids read at different levels when warm instead of cool. Locate the power steering fluid reservoir. It is located under the hood of your Toyota Corolla on the passenger side next to the engine. The reservoir is connected to the power steering pump. Remove the cap and use the dipstick to check the fluid level. For the most accurate reading, clean off the dipstick with a rag first, then re-insert and remove for the fresh reading. Add automatic transmission fluid or power steering fluid approved for use in Toyota vehicles if the fluid is below the low level mark. Add fluid gradually to keep for over filling the reservoir.
this will help.thanks.

power steering pumps either work or don't, so the other part that does cause this issue and will be intermittent in nature is the steering rack, u have a leaky assist piston seal.

If yours is a 2003, you might look at changing the power steering fluid. I have a 97 Accord that would start jerking the steering wheel back and forth after it had warmed up. Replaced axles and didn't work, but replaced the fluid and it hasn't happened since. Wouldn't be hard or expensive to try. Just be sure to use the right fluid.

does it have power steering ??chances are it doesnt if you cant find a resevoir