For your information, bruxism involves teeth grinding/clenching, so no, it isn't self-harm. Biting one's cheeks, on the other hand, could be considered as self-harm.
As for dating someone with self-inflicted scars, I wouldn't be turned off by them. However, I would be influenced by the person's mental state and the outlook they have on life at present. I cut myself on and off for 2 years in my early teens, but now I regret the fact, and feel embarrassed even thinking about it. I realise it was a very stupid thing to do, and wouldn't even consider doing it again. Many other former self-harmers are the same. Hence I would have no problem dating someone like this.
But I would probably find it difficult going out with somebody who is depressed. Depression, and eating disorders and various other mental illness, take over a person's mind, change how they think, make them focus excessively on themselves and their problems. It's not their fault, but it does make it difficult being around them; sometimes it feels like you're talking to a brick wall, because your voice of reason just doesn't seem to register. It is very emotionally draining being with someone like that. Some people could handle this, try to change them, but realistically, someone can only change if they are ready to and want to themselves. I've been on both sides of such a relationship, and, honestly, wouldn't want to do it again.