Chavelier, you think in a very romantic and pure way. I do too.
You know you only have good intentions to befriend women - so go for it! Your Freudian slips, if you even had any, would show nothing wrong.
Just go there, go up to someone and just talk. Like, if she's sitting down on the staircase - in my eccentric and wierd way, I would probably go up to her and say, "You know, there's a perfectly good chair over there" - complete with a point to said chair.
Now that's my way of starting conversations, and it's a way I like and am comfortable with. Common territory is very much needed for any conversation - so why not start it with something both of you can see?
Now the only think preventing me doing that to everything is a natural shyness. People have it. But it doesn't hurt going up and talking to someone - and keeping it light means you can break away quickly.
But don't go purposely out of your way to sit next to her in the lecture hall... that'd be kinda wierd. Do that when you're friends.
I'd only ask a girl out to coffee if I tried the above environment-based conversation and then it went on for a while and I had to leave. I'd probably say something but I have no idea what to say... so I'd have to make it up on the spot. Something like, I have to go now - it was great talking to you. Hey listen, I know this great coffee place - want to go chat some more there after such and such?
Go there, and just be yourself. You lapse into your natural shell anyway - just go there, be interesting, be friendly and all will work out. You'll find someone. Like at the UNSW Info Day, first girl I talked to - we got onto a twenty minute conversation. Me, the guy with perennial girl problems. It was... shocking, to say the least.
If I can do it - sure as hell you can.