What is the whining noise when I start my Corolla T Sport from cold?

A high pitched whine like that indicates a bad bearing. Water pump, power steering pump, accessory belt slipping, alternator bearing, or in the worst case scenerio, Transmission pump.

Slipping serpentine belt. Spray it with belt conditioner.

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check the belt tensioner.

The noise you're hearing is the electric fuel pump in your fuel tank.

This could be anything from the pump in the transmission to the alternator bearings/bushings to the power steering pump. Could be an idler pulley bearing, but I doubt it.
Hint: If the noise is very high-pitched, check the alternator. If it is a lower, medium-high-pitch sound that changes some when you turn the wheels, check the power steering pump. If it is almost like a moan or a "Whooo" sound, that's the transmission pump.

power steering pump is either low or needs replaced

Your water pump is going out

Have the tranny serviced with fluid change and filter, if noise it still there it is the torque converter . Have the dealer fix it ,and have it on paper with dealer invoice when it is done. Do all this before warranty expires. IT IS NOT USUAL for any car, Toyota included, to have tranny whine!!!!!

Sounds like a dry/failing bearing or power steering pump noise. You first need to try to track down where the noise is originating, and the most likely place is at the front of the engine.
I would start by listening with the hood up while parked with the engine running and the noise present, and seeing if it is coming from there. If so, the most likely cause is something connected to the belt - e.g. AC compressor clutch, tensioner, power steering pump, or generator. Can also be caused by the belt just slipping (warn out belt, tensioner not working properly).

Your transmission is going to fail sooner or later, the whining noise you hear is the the clutch packs in the transmission going out.