Does impurity transfer to other places in the house as a result of my daughter trailing over impure areas with her urine, and what is the ruling on the places her wet feet passed over? And does the vacuum cleaner tank, which drew in the impure water, become pure by passing pure water through it? And is the entire house considered impure as a result of this?

Your small daughter's walking in a certain area is not enough to rule her socks impure based merely on doubt, as the fundamental principle for anything is purity. As for a vacuum cleaner that sucks up water that has been altered by impurity, any part of it that the water touches becomes impure. It is not purified by dust, but rather by purifying its interior through washing. Wiping is sufficient if it is polished. Anything you doubt has become impure is considered pure according to the fundamental principle, but anything you are certain has become impure must be purified.