Does madhy—which touched the car seat, on which there is a carpet that is neither light nor heavy—transfer to the clothing, given the presence of wetness from the water used for istinja?

The basic principle concerning things is purity, and what was mentioned is a form of waswasa (whispering of Satan/obsessive doubt), so avoid it. Doubt about the presence of impurity, when one is certain of prior purity, does not invalidate the prayer. For doubt does not remove certainty, and there is no need to purify with sprinkling (nadh) that about which one doubts its impurity.