Is that which the worshippers stepped on with their shoes after exiting the restrooms onto the mosque carpet considered pure, and how can the carpet at home be purified if a child urinates on it?

The restrooms should be outside the mosque, or at least their doors should be outside the mosque. It is not necessarily the case that whoever exits them will have impure feet or shoes; the basic principle is that a Muslim takes precautions against impurity and removes it with ritually pure water.

If impurity gets on the carpet or rugs, it must be removed and purified by pouring water over it and squeezing it until it disappears, just as the Companions did with the Bedouin's urine. This ruling applies equally to the mosque, the home, and other places.

However, if there is merely doubt about impurity having occurred, sprinkling (nadh) the doubtful spot with ritually pure water is sufficient. And if what touched the place, body, or clothing is doubtful in its impurity, then it is not obligatory to wash it or sprinkle it, because the basic principle is purity.

In summary, if impurity is confirmed to have occurred, it is obligatory to remove it and purify it with water. If there is doubt about its occurrence, sprinkle the doubtful spot. If there is doubt about the impurity of what has touched one, then nothing is required, for the basic principle regarding things is purity.