Is the impurity considered to have transferred to the wet foot upon stepping on a spot likely to be impure on a bed where a child urinated, then walking a few steps on another mat before washing the foot?

The fundamental principle is the persistence of purity and the non-transfer of impurity. If you doubt the impurity of something, its original state is purity, and certainty is not dispelled by doubt. Therefore, your foot is considered pure due to the lack of certainty that it was afflicted with impurity, and consequently, the second rug you walked on does not become impure.