Warren, the same thing happened to me 11 months ago, don't sweat it. I've shed a few tears for girls too, but don't let it get you down. You have to look proud and confident, not sad and dopey. If you truly want to "Get to know her" then you need to act like her "no" it didn't affect in the way it did. You could tell her that it hurt you as a friend because you just want to get to know her and hang out with her more, but don't let her know that you were really hurt 'in that way' when she said that.
Don't ask her why she doesn't like you. That will just be badgering her and she'll most definitely start to avoid you. Be prepared, she may do that anyway, but before she does explain that you just want to be friends and get to know her. You're in a tail spin right now, but you haven't hit the deck yet. You've just got to pull yourself straight and start flying again.
Actually, it would probably be a good idea to tell her that it hurt you as a friend, because that friend of hers that told you probably told russian chic your reaction and that 'hurt as a friend' would explain your reaction. And do remember that she is being COMPLETELY and TOTALLY nice and considerate about it. If she wasn't she wouldn't have told you "no" right away. The similar situation I was in was made 10 times worse because she never just said "no, I don't want to go out." So she obviously cares about your feelings, but don't take that caring to a dillusional extreme.
My computer is leaking water profusely, so I can't type much more. If it's as bad as I think it is then I won't be able to use my computer for a week or two

. I'll just have to get some silicone and fill in the hole, I hope that works!