The reason why I’d assume Nitorcis would shout soon after opening the door is because she could only wait for so long in the corridor before Battler would get suspicious. I’d expect it’d take some ten seconds to get to the door, a moment to unlock it, and then some fifteen seconds for Ritsuka to go upstairs quietly enough to be unnoticed. If Nitocris just waited in the corridor for half a minute before doing anything, I imagine Battler would’ve started wondering about what’s up with her. If they were already finished with examining the bedroom, Battler would’ve followed sooner. If they weren’t done already, Battler would wonder why she’s going outside and spending any extended amount of time there. The narrative didn’t mention Nitocris saying anything to Battler about going ahead to the room they agreed to meet on, which could’ve explained this.
Well, if Robin was murdered, it more or less has to be Mashu who killed him. Let’s try a couple of things. She had no reason to blame Robin, but she did so anyway. She was acting irrationally. She needed to blame someone, and Robin was there. Alternatively, the whole murder mystery thing was Mashu’s idea, and thus Robin blamed her for Ritsuka’s death. Robin attacked her, and Mashu killed him in excessive self-defense. Or perhaps Robin did not attack her, but blamed her for Ritsuka’s death, and Mashu wanted to shut him up because she couldn’t take it. She didn’t really intend to kill Robin, it just happened.
Ritsuka’s corpse was not moved after death? Mashu carried him away, but he wasn’t actually dead until she’d already brought him outside the mansion grounds. Or perhaps Ritsuka walked out of the mansion grounds himself, before finally succumbing to his wounds.