I am probably blind but I did not see a requirement for the District to be Monster free for a Hero to use a City Action.
Can a City Action be performed if the District has Monsters in it?.. feels kinda weird that you could. We allowed it during our game but it feels odd.. but with such limited time due to the Apocalypse Track this may be correct.
Thematically, I'd say the monsters are generally patrolling the main area of the building and players can scurry into side rooms and storage areas, med bays etc to find items or do whatever. Monsters only attack when the OverMind of the Sin player tells them to, just getting some swipe in when players expose their location to exit the area.