In the first scene, the sword used by the yakuza is actually a wakisashi, shorter than the regular katana. The scabbard is also worn in the wrong orientation, you would usually wear it so that the edge faces upwards, not downwards.
In the hospital room, a ventilator is heard but the patient is not ventilated (aka artificial/assisted respiration) but only has nasal prongs on.
The officer should relatively easily spit out or push out with his tongue the piece of cloth in his mouth as it is not taped over his mouth.
There is no reason for a character to remove his hair cap while on a murder site: it will increase the risk of hair falling on the floor.