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I am assuming you have the "Variable" power steering system, otherwise, these types of symptoms would not happen with the power steering.
Of course, the proper way to debug this type of problem is to connect the car to a scan tool that can read the parameters (signals) from the ABS system. ALL of these symptoms are coming from this system (ABS & Traction control). Here is a basic diagram showing how the Electronic Brake Controller sends a "speed signal" to the speedometer and the power steering system.
What YOU can do, is inspect the wiring around EACH of the wheels and look for any loose connections (connectors & wires) around the wheel speed sensor connectors. Here are some drawings of the front and rear wheel speed sensors.
Front (item 3 is the sensor)
Rear (item 1 again. Item 3 is the differential housing)
If you don't find anything obvious, you will most likely have to have it diagnosed with a scan tool that will tell you which sensor (or sensors) is failing. Otherwise, you would just be guessing and throwing parts and money at the problem.
I hope this helps. Please let me know if you have more questions.
Take Care, Greg A.