Just picked up a 2003 S-Type R and am having the same range of problems described here. The low speed "jolts"...the transmission occasionally feels like it is disengaging, especially after moderate braking. When you get back on the gas it jerks back into gear with a thump. It has a random element to it and cannot be made to do it every time.
I have Pirellis on the car and one of them developed a sidewall "blister" after a 1000 mile run. Maybe this car is too heavy for those tyres?
Thanks for the tip. I am off to the dealer to talk to them about the the transmission reprogramming.