Although the skin problem idea is somewhat plausible, I still don't think that's the reason why he uses a full-face mask. During the hospital kidnapping scenes, he only covered the bottom half of his face with a surgical mask to simply disguise himself and wore a pair of glasses to impersonate the real doctor. As obvious as it sounds, I think he wears the mask literally to disguise himself and to give a creepy skin-crawling effect of "no face". I don't think there's any other particular deep meaning behind it. As for the older brother, he is very suspicious (and definitely up to no good tbh) but I feel like with kdramas like this, the writers are probably purposely trying to mislead us?? I have a feeling it's just a completely different character with obsessive psychotic intentions.