quesosoup:
"The apple doesn't fall too far from the tree."
Now that you mentioned this, I thought of something. We were shown in the earlier episodes that Do Do Hee's parents were killed in an accident. However, we are not given many details about the accident. It also seemed like Madam Ju felt guilty about it. Noh Seok Min also mentioned in one of the last episodes something about how dare Madam Ju judge him. Let's also add that the scar guy seems afraid of someone coming after him.
I do find your theory about the scar guy being Seok Min's dad plausible. We are never shown how or when he died. I was also thinking it could be Do Do Hee's dad, but then why would he try to kill her?
I don't know how all of this is related or if it's even related, but it seems, at least to me, that we have been getting small clues about who it is.
Noh Seok Min: Most likely not the killer.
Noh Do Kyung: Psycho, but it would be too obvious. However, I do feel like the killer is connected to him in some way.
Joo Seok Hoon: Again, it would be too obvious to make him the killer. Instead, I think he plays a big role in Do Do Hee and Jung Koo Won's past.
Jin Ga Young: Most definitely connected to their past life and probably one of the direct or indirect reasons he became a demon in the first place
Kim Se Ra: She knows something.