When the ex crown prince told Geon that Samyeong isn't himself anymore AFTER his wife and child are dead. The panel showing he cut his hair implied that he went mad (i.e., denying the truth) bcs being a good son he is, he wouldn't do that. He is just trying to be happy (before he met Geon again). I think the panels are coherent and understandable on how simple it is.
I mean, that's how grief works... It was kinda implied that the child couldn't survive on her own alone (I think I kinda remember characters saying that ????) but Sam-myeong is still in denial and says his daugther is "lost" somewhere. It's kinda confusing because Chang-geon also tell the guard to find her and kill her, but I guess it's just cause he believes Sam-myeong lol
Unless she's actually alive and the author is playing with us
I am very lost. How did he turn to be a king when he was so below his brothers? I doubt his dad choose him to be one. How the older brother let him take his place just like that? Showing us his past just to justify the reason he behaves like that doesn't help. He turned to be the person that he hated the most. I understand why he become upset with his brothers and father but I don't think ir justify the way he treated the only person that treated him like a human being by abusing him like that. Ww are what we choose to be. He wanted to be loved but he chose the wrong way of doing it I believe.
I think you need to reread to understand better.
The father died (the first chapter says bc of old age, so we don't know if it was CG) and CG killed his brothers (again, the first chapter says they died bc of "unknown reasons") in order to obtain the throne.
He may be born from a concubine, but he is still in the royal line. He still was a prince, so, since there wasn't any other one in the direct lineage of the king, CG got it.
About the other things you said, I agree lol

Eh? I'm so confused.. the short hair mc scene is from the past or present? I'm so confused with the whole chap and the long haired one in the end of the chap is also mc? So then if he went crazy back then how's he's sane now?