Is the prayer of one who prays on a prayer rug placed in an impure location, with the rug itself being pure, accepted? And does the ablution of one who places his foot on a spot where a person walked with a dry, impure foot become nullified?

The default state of things is purity until proven otherwise with certainty. Therefore, streets and feet are, by default, pure. Praying in the middle of the road (on the roadside) is disliked due to inconveniencing passersby and a lack of focus (khushu'), but the prayer itself is valid.

If a person prays on an impure spot, their prayer is valid as long as the specific area of contact with their body is pure. Likewise, touching impurities does not invalidate ablution (wudu'), and dry impurity does not transfer if the object touching it is dry.

One must be wary of obsessive doubts (waswasah) concerning religious rulings and strive to learn them, alongside repenting for having missed prayers and making them up.