Agreed that having her on Facebook could provide some helpful info on what she likes and whether they have things or interests in common, things like that. And info on her relationship status would definitely be helpful as well, that's true.
But as to whether it's a good way to initiate contact... it may be, but seems to me it rather depends on the girl, and more likely than not the answer is no?
Not sure I have any helpful advice of my own - the last crush I had half a year or so ago suddenly left the country (okay, by American standards she'd be less than a state away, but in here it's the next country ) before I got past hesitating how to make a move, so I guess if anything my advice would be not to hesitate too long.