Pretty sure he wanted to test him to see if Kwak was suspecting him of murder. And I think the nail is real since we know Dangwoo keeps the clothing of peoplehes killed, so why not accessories? If Kwak hadn’t taken the nail and hadn’t lied about taking it, Sangwoo might have let him go, hence why he covered his injuries. But since he know knows he’s now being suspected by someone higher up, he’s gotta decide what to do. If he lets Kwak go, Sangwoo will remain a suspect. If he kills him, he’ll buy himself some time but will also make the other officer even more eager to catch him. And since Sangwoo is prett unhinged right now, I think he will kill Kwak. But like you said, we’ll see.
Maybe he plans to play it off? I mean, since everything just now was planned as well? Or he might just kill him and drag him to the basement, guess we'll see? I mean, he literally gave Kwak the evidence, maybe it's fake? Or maybe he wanted to kill someone to let off the pressure? I wonder how he'd play off the blood?