Just because Dohu tells himself he doesn't care what others think, it doesn't mean it's true. That's just what he tells himself.
He's an expert in tricking himself into believing he's not hurting when he is, just like what he's doing now by preparing himself for when Juheon leaves him. He already knows he'll be hurt by that, but he pretends that he won't.
In my opinion, it's more like he's the type to lie to himself that he's not hurting when he is. It's his defence mechanism because he is a little different from his peers (he is slightly on the autism spectrum) so his whole life is them misunderstanding him and insulting him.
But his defence mechanism only numbs his emotions when people are insulting him. He's not used to hearing the people he cares about get insulted (probably because he has so few friends). That's why he will still get angry for ML.
I really don't get whats Dohu personality like.. I mean in early chaps they describe him as someone who don't feel emotion but then he get angry with the annoying (sunbae) prick. Now they are back potraying him as someone so aloof to the point he don't give a fuck to what other feel or think. Arghh so confusing..