Do socks become ritually impure because of wearing shoes that a dog had taken and played with in the street for a day or two, then were washed once with water, and then worn a month later with socks on?

If the shoe is contaminated with dog saliva, it must be washed seven times, one of them with earth, according to the Shafi'is and Hanbalis. Soap and similar cleansers can be used in place of earth. However, if it is not certain that it has been contaminated with dog saliva, the original state is purity, and it does not need to be purified. If it is washed only once, while being certain of its contamination with dog saliva, that is insufficient. As for the transfer of impurity to socks, if both the footwear and the socks are dry, then the impurity does not transfer.