My take on family entering the apartment IMO is, they first were allowed to enter, then ushered out and forced to sit in the lobby. Pure speculation, upon first finding Julia YPD didn't know they had a possible homicide on their hands. They notified the family early, to report the death, which is customary.
In regards to the family showing up at Julia's apartment, KT reacted like a caring, concerned and shocked parent would in keeping with such tragic news. KT and JN (twin) trying to storm the bedroom/bath area and being stopped by YPD seems in keeping with as well.
As for KT looking at JT and that's how he knew, IDK what to think. I would guess if someone was called to a adult child's apartment who was reported missing by roommates and LE and an ambulance were at the scene, a person would know something very tragic was happening. Someone was injured in someway or having a medical problem, at the very least. But how JT could know it was Julia's death by just looking at KT, in her distraught state, again IDK. How could you just know it was death? LE and an ambulance will show up if a person is: unconscious, beaten, raped or out of their mind. Now, this is KT's statement, not JT's so without hearing from JT, I find it difficult to know for sure if JT agrees with KT's statement.