Are female soldiers allowed in the male barracks?

That would depend on the unit commander and the situation. In basic/AIT females were excluded from the male barracks but at permanent party assignments there were no such restrictions as long as nothing was occurring that shocked the sensibilities of roommates etc.

How, Why, When? Too vague here PV2- During duty hours, same unit, doors open- probably. After duty hours, same unit open doors, maybe. After lights out- NO. In my day no female was allowed pass CQ desk, but we didn't have combined gender training then either. The Command undoubtedly has a barracks SOP Posted somewhere- ask you NCO's

SSG Jeremy Sharp makes a good point depends on the unit and commander; most time as long as you are in training then it shouldn’t be a problem. I been in females barracks all the time that was permanent party