(Answer contains spoilers from the novel)
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That's because he is not naturally white-haired (and his white hair is a relevant clue for what happens later in the novel with the apocalypse, too). Euijae's white hair is a consequence of what happened in the rift dungeon (only he doesn't remember until almost the last 50 chapters of the novel). He has always been naturally black-haired. And since he lost his memory of many things in the past, dyeing it's the best way to not get attention on his persona. This way, he just looks like your "average" young man, too, since only a few people from the first two timelines knew his face aside Lee Sayoung (and most of them do not know in the third/actual timeline).
The mask is probably because even if he doesn't use it, people will eventually start noticing his skills and abilities, which will lead to being recognized not only as J, but also as "J's face". He wants to separate his entity of "Hero J" from "Cha Euijae", and for that, he has to hide his face. In the past nobody saw him as a human, only as the hero, and that's one of the reasons he ended that way in the first two timelines. So the mask is more like a symbolic reason, especially because he will end up coming back officially as J too soon, and that lets him have some freedom as not being recognized as the "part-timer of the hangover soup shop".
Current timeliness is 3rd—during chapter 300+ it was vaguely mentioned that 'How many worlds did you destroy' to the 'White hole' aka the apocalypse/end but anyway in the memorial dungeon (mid 240+ till 300 and also parts near the ending of the chapters we'll get to see Euijae and Sayoung from the other timelines(the ones that was alive in their timeline)
Currently? No but before part 2 ends (maybe around 180+ chapters in the novel) they will meet 2nd timeline Lee Sayoung cuz he would(?) fight with current sayoung for the body but in the end they did merge but I think Euijae knows because student(the girl i forgot the name) did mention fragments etc² but Im not sure

I forgot. I know as J he had black hair but originally he’s a natural whitehead. So why does he need to dye it? Was it cuz of the rift that changed his hair color and having white hair would look strange in today’s society? Also why does he cover his face? I feel like no one would recognize him if he had white hair and no mask