You get equipment from defeating enemies and looting chests in vanguard battles, finishing the battles, and vanguard arena battles. Or you can buy them on the overview map from the caravan, when it appears.
The icon for Kurohime's event appears on the country overview map in Dorgania when you finished all previous events with Kurohime, Ryen fucked Lady Akira at the brothel, and you prepare for the war with Dorgania. (Going to war isn't even possible yet)