Do the feet and furnishings of a house become impure due to children's urine that is wet on them, or due to worshippers passing with wet or moist feet from wudu on the mosque carpets?

It appears you are experiencing waswasah (whispering of Satan/obsessive thoughts), and it is necessary to disregard these waswasah and pay no attention to them, because giving in to them is a cause of evil. The default state is purity. Therefore, as long as there is no certainty that impurity has transferred, the original state of purity remains. Accordingly, as long as there is no certainty that the child's foot has become impure with urine or anything else, the original state of purity remains. As for the transfer of impurity from an impure foot to the prayer rug—assuming the foot was impure—if both the foot and the rug were dry, the impurity would not transfer. If one of them was wet, there is a difference of opinion on the matter.