Is it permissible to pray on household carpets that were stained with children's urine a year ago, and the spot of impurity has since been forgotten, and is it permissible to step on it with a wet foot?

For the validity of prayer, the place where it is performed must be pure, such as a ritually impure prayer rug. It is not permissible to pray on it until it is purified. If a garment or place becomes ritually impure and the exact location of the impurity is unknown, then both must be washed entirely. As for treading on a ritually impure prayer rug with a wet foot, the principle is to avoid it. However, if it occurs, the foot is purified by passing it over a pure place, and washing it is not obligatory, as this is something that is pardoned. This is supported by the hadith of Umm Salamah, where the Prophet, peace and blessings be upon him, said about a woman's garment trailing in an impure place: "What comes after it purifies it." And the hadith of Abdullah ibn Mas'ud: "We were with the Messenger of Allah, peace and blessings be upon him, and we did not perform ablution (wash our feet) from treading (on impurities)."