He was "hurt" by Frostmourne. It's implied that Frostmourne's ability to steal souls is a passive trait if you will. Essentially being in contact with the blade for long periods of time will cause it to claim your soul as it's stated that Arthas' soul was the first one Frostmourne had claimed despite never being stricken by the blade directly.
It could be tied to the will of the person? A dead person has no strength to resist the blade's influence, whereas a living person won't be immediately corrupted. Somebody killed by the blade would have their soul immediately claimed as they no longer have the strength to resist the blade's magic.
But the second part is totally accurate. There's no reason to suggest that having white hair is a trait specific to Arthas in the slightest. It's just something that happens. Seems like its an effect of being "Death Knighted" that your hair color changes, and blonde must change to white specifically.