What are the proofs of those who consider alcohol to be impure, and does this entail the impurity of alcoholic perfumes? If so, is the prayer of one who has perfumed himself with them valid after the alcohol has evaporated and only the scent remains?

Alcohol is impure (najis) and it is not permissible to perfume oneself with it. What some scholars have concluded through their ijtihad (independent reasoning) is contrary to the view of the majority. If alcohol touches clothing or the body, the affected area becomes impure and must be washed with water, regardless of whether its trace remains visible or has dried.