Are the waters discharged from the sewer opening, which have mixed with rainwater in the street, considered impure, and what is the ruling on shoes touching them and entering the house and car with them?

The default state of water, land, and clothing is purity. Therefore, if you doubt the nature of flowing water, rule it to be pure. If you are certain about the presence of impurity in rainwater, consider it among the excused impurities, provided the impurity is not overwhelming or standing, especially if it is the impurity of roads that has splattered and become indistinct. It is not necessary to purify the car floor or shoes unless you pray on them, and you must avoid the obsessive whispers of tracking impurities.