During the sword fight scene, when Akiko first appears on the roof, she is wearing ankle high boots with a short spike heel. During the fight, she is wearing similar boots, but the heel is now flat.
Aiko comes out on the roof with shoes with kitten heels, but wears boots when she fights.
When Patterson is freezing the safe lock to crack it she holds out her bare hand to catch any spillage of the freezing solution. No scientist would use their bare hand to catch dry ice or any other super freezing substance.
Rich hands Aiko a Nihonto, a live sword, with hilt correctly facing to the left, but with the blade facing Aiko. It is considered an insult and compels the receiving person to fight if one were to hand a Japanese blade with the blade toward them. Rich might be given latitude as gaijin, a foreigner, but Aiko certainly wouldn't receive it graciously.