Is a dog's bite considered like other impurities in terms of purification, or is it like a dog licking a vessel, requiring washing seven times, one of them with earth? And do the skirt and leg become pure by washing them once after a dog's bite, given that there is no recollection of saliva reaching them?

As long as there is doubt about the wetness from the dog's mouth reaching the leg from behind the skirt, the ruling for the leg is purity. As for the bitten area of the skirt, the Maliki school of thought holds that a dog's saliva is pure, and that the command to wash a vessel licked by a dog is for recommendation as an act of worship, and it is not recommended to wash anything other than vessels, such as food or clothes. The person afflicted with obsessive doubts (al-waswasi) can adopt the Maliki view on this issue and consider it pure.