What is the ruling on one who steps on dog urine and then walks on the ground without rubbing his foot, and is it purified by that? And what is the ruling if the urine was dry on the shoe and he walked on it?

If the foot steps on non-dry dog urine, it must be washed, and merely rubbing it is not sufficient. As for the sandal, if dog urine gets on it, it must also be washed if it is not rubbed. However, rubbing it is sufficient to purify it according to many scholars, and they do not differentiate in this regard between the impurity of a dog and other impurities.