"Anyway, so how much attention and care do you put into how you look? Do you, like, put a load of effort into it, or do you just occasionally cut your hair and wash yourself and call it a day."
I used to not, but I spend a fair amount; not too much but I would call it just a bit either. Somewhere in the middle.
"Also, some people care only for appearances. I'm just wondering how you people would feel if someone came up to you and complimented you on your looks? "
I would take it. I don't see what the contempt and scorn for judging based on appearances is nowadays; it still is part of someone.
"What if a friend or someone was friends with you because they found you looked good. What if the opposite even? What if they didn't like the way you looked, and were only friends with you because of your personality? "
Friends are friends. If I trust them enough to be friends, I wouldn't question their motives.
"What about someone else? Do you feel it is "okay" for a person to like another because of their appearance? Even if it's mutual or the receiver enjoys the attention? Do you think it will breed a strong relationship? Could it even be called a relationship."
In fact, I do believe it's okay. As I said before, it's a part of someone! I don't see how enjoying the attention would make it any better or worse. I don't think it'll breed a very strong relationship, especially if the people in question can't stand each other, but our minds do weird things to our perception when we're with attractive people, just saying.
Additional notes: Recently, I find that I've been seeing owns own appearance as a form of vanity(well, with a more positive connotation, but I can't think of a proper word right now), by which I mean ones own satisfaction. Really, I think it's enough in terms of appearance if you find yourself acceptable, though if others are to judge... That could interfere with it a bit, but I do believe that is a natural thing. Appearance really has a lot of far reaching effect and I don't think it's fair to isolate it on its own.