Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
If there is urine or feces or any kind of impurity on the carpet, you must wash it. If you walked with wet feet on the same spot, then your feet has contracted impurity, and your prayer is not valid unless you remove the same.
In responding to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
If your body or clothes or spots where you pray or live has drops or traces of any impurity such as urine, feces, etc., on them, then it is sufficient for you to wash it thoroughly with water. However, using some kind of soap or detergent with water is considered even better, if you can afford it, for Islam teaches us to do what we do as best as we can. Washing with soap/detergent is definitely cleaner.
The same applies to washing sperm or pre-sperm fluids. The Prophet (peace be upon him) said, “When Allah provides better means for you, conduct yourself appropriately.”
If there is urine or feces or any kind of impurity on the carpet, you must wash it. If you walked with wet feet on the same spot, then your feet has contracted impurity, and your prayer is not valid unless you remove the same.
If you had prayed without removing the impurity, then your prayer is rendered null and void, and you must repeat your prayer, unless you are sure you have removed it or it got itself cleaned by walking on clean sand/earth, etc. Clean sand or earth can be considered as a purifier, but it is still considered preferable to wash your feet when in doubt. Islam orders us to do things as efficiently and properly as possible.
We must never be slack or relaxed about the rules of impurity. For the Prophet (peace be upon him) said, “Prayer is the key to paradise; the key to prayer is purification.” Purification is considered an essential pre-requisite or essential integral of prayer, without which it is not considered as valid.
If you are sure that you have walked with your feet, which had traces of urine or any kind of impurity, on the carpets of the mosque, then it is considered your Islamic duty to make sure that they are cleaned.
Likewise, your family members who have done the same must also bear responsibility for cleaning them. We are accountable for our actions. Those prayers that you have performed with impurity on your feet must be repeated.
If the child has urinated on the cloth, then it is impure. And you ought to wash it.
However, if people stepped on it by accident and then they have walked on another dry surface, then they need not worry as it should be like someone walking on filth on the road with his shoes, and then walking on the dry path later; the latter should help clean it.
It is always good in case of doubt to clean the body parts or clothes that have come in contact with the impurities.
Almighty Allah knows best.