How should one deal with a non-Muslim girl who frequently visits a Muslim home after her family acquired a dog, and is it obligatory to purify her hands and clothes seven times, one of which is with earth, upon each visit?

The correct view is that the dog is ritually impure with a severe impurity (najāsa mughallazha). However, the default state of the girl's body and clothes is purity, unless a specific impurity is known for certain. There is no obligation to purify what this girl touches unless you are certain of the impurity. In that case, it is obligatory to wash whatever came into contact with it, provided one or both were wet, because something that has become impure will transmit impurity to whatever it touches, even if the place of impurity dried previously.