About your original question: there's an informal old-fashioned and seldom-used term for an event where women are supposed to invite men to dance. It's called a "Sadie Hawkins dance." I doubt many young people today would even know what that is. The only formal word for dancers I know of is the one used in ballet. Ballet dancers are know as "ballerinas."

At a formal dance or ball, a dance where the women ask the men is called a ladies' choice. It doesn't need to be a slow dance though. It could be any couple's dance.