What causes the brakes to pulsate and the wheel to shakes when stopping?

Traction Control taking control...feathering application of the brakes to smooth the stop...only ain't all that smooth~

Could be old fashioned uneven wear on the rotors requiring a job to the machine shop to put true again~

How much meat left on the brake pads? 3mm...2mm.---0mm??

Your rotors either need to be machined (if enough thickness remains) or replaced. Since you feel the vibration through the steering wheel, it is at least your front rotors, if not all four.