Did the clothes of the woman performing the ritual become impure when she sat in the car while wet after her bath, in the same spot where her five-year-old nephew had sat after urinating and not cleaning himself, and must she repeat the Tawaf and Sa'i because of that?

This confusion is due to waswas (obsessive whispering), and you must ignore these whispers and doubts. The default status of things is purity, and impurity is not declared to have transferred except with absolute certainty. Based on this, your tawaf (definition) (circumambulation) is valid, and your body, clothes, and the car seat are deemed pure, as there is no certainty of their contamination. We warn you against giving in to waswas, as it opens a great door to evil.