Does the bottom of a shoe become pure if it is stained with an impurity and then walked with on a pure place that is not dust, such as carpet, marble, or the street?

You must ignore and combat obsessive thoughts (waswas). As for the purity of the bottom of a shoe, dust purifies it. However, things other than dust do not. But dry impurities do not transfer. You must verify if the shoe has been afflicted with impurity, as the original state is purity when in doubt. Many scholars consider a small amount of impurity to be excusable.