I have Volvo S60, 2002. Since some time, about 10k miles, I had increased steering problem. During fast acceleration car has pull to the right side so I had to turn my steering to the left to correct it. Car has now 167k miles. Had changed recently controls arm, ball joint left side, spring seat and upper column bearings, subframe bushing. About 50k miles ago had been changed tie rod ends. Now after last repair - subframe bushing change the problem is less visible. Before the driving was somewhat danger. After hard acceleration, when I had taken my foot from the pedal gas, the steering wheel so repositioned that the car made sharp side turn. Now it is very visible when I am starting and for example (this is not how I am driving) the steering wheel is not kept it is turning 45 degree to the right. When is kept I have turn it a little to the left to correct the ride. Every thing is visible and notisible during fast acceleration. Car had done alignment and also the mechanics did not find any think required work. Also during normal driving it gives feeling that it is unstable, car is moving a little from left to right and right to left. It was very easy to loose control on icy rods even when car equipped with winter tires.
What can cause the problem?

 

That pull to the left you are describing under hard acceleration is called torque steer. It is what happens when you have a lot of power going through the front wheels of a car and the chassis is not quite up to the job. Its a problem on the 2.4T, T5, And T5R although not so bad on the 2.4T due to the light pressure turbo and the smoother delivery of power. There is not a lot you can do about it unfortunately apart from ease of the go faster pedal.
If i accelerate fast in my car (2.4T) and do not hold the steering wheel it will pull in first and a bit in second gear. (just dont release the steering wheel under hard acceleration)

As for the wobbly steering you are talking about. Have you checked your wishbone bushes.? These are a known weak point and will cause bad steering.
The steering on the S60 is not the greatest in the first place to be honest as it gives very little feed back and does not inspire you to really drive fast in to corners And also with the 225 or 235 wide tyres they do tend to tram line which could also be your problem as well as tyre brand. Pirelli P6000 tyres are not a good choice for this car.

 

Befor the subframe bushing has been changed and the control arms, the car turn opsite direction (towards left) during braking than as during acceleration (toward rigth). Now car stability looks ok during breaking. I am suspectiong that something lase can be weare out like the stering rack. Could it be?