I wouldn't invite her anywhere outside as it's counter-productive. I really think you're over-thinking this because of the amount of info you've learned through the PUA scene. This is a really simple situation. You met a girl. She thinks you think she's single. She's giving you lots of IOIs. Like with any girl you need to make your move or she'll move on and find someone else. Invite her over to watch a film or something and then make your move. The worst that can happen is she politely refuses. Taking her to sporting events would be a big mistake and you aren't gonna share your first kiss infront of thousands of people and definitely won't be able to escalate from there. Close her first and then take her out. Girls that age are rarely committed to their boyfriends. Find out if that's the case. Obviously don't ask, just make your move!

UPDATE

I just want to apologize to the pua community, y'all were right. At a party today and thr girl with the bf was hooking up and front to front grinding with her RA/my teammate, shoulda listened to ya, but I guess I don't want to be with a cheater (find some bright in this darkness). Anyway, touched her hair and hugged her, so at least I got a bit of kino in with no result.

go learn to cold approach, you're needy and you don't try to sleep with enough women, start trying to sleep with them, instead of trying to get them to sleep with you

also, don't take needy as some bad thing that you have to defend yourself against, no one even knows you in person on this forum anyways, there is no need to convince anyone that you are this, or you are not this, it's simply a form of behavior, think of it like this, the opposite of needy is indifference, now if you are indifferent to a girl, there are a few things that happen
1) you don't care one way or another if you have sex with her
2) you don't care about if she knows you find her physically attractive or not, cause you can take it or leave it, makes little difference
3) you don't care about what she thinks about you, or what she will think
4) you are not afraid to say and do, what ever you want around her, that includes trying to make out with her, cause it doesn't matter to you if she is in your life, or out of it
5) you view her just as a person, rather then someone special

now, if you care = needy, don't take this as someone ripping on you when they say you are needy, every guy and his damn dog gets needy, it's a pattern of behavior that you will start to exibit when you think you have no other choice but to connect, or when you think someone has something that you want, and it is important, learn to spot it, and when you are getting needy, cut yourself off from the source that is causing it, and meet other girls to shift your focus so you don't get too emotionally invested, when you feel like you could give a fuck one way or the other, you are probably good to go take another shot at the source of the prior neediness

also, don't confuse not caring, with not being aware of your social surrounding and how you are making people feel, that is not the point, the point is simply to be indifferent to the girl entirely, you can take it, you can leave it, you don't care, you can hit on her, it isn't a big deal, as long as it interests you, you will stick around, but if it isn't of interest anymore and you are being abused or something, you don't have to take that shit, you don't really need to be around her, she's just not that much of a big deal, not a single fuck is given

really take concious notice of this, and actually become indifferent, rather then acting indifferent, actually be indifferent, have enough shit going on in your life that you don't give a shit one way or another what happens, and you just do what you want, so figure out what you want, and start going for it, don't try to get it to come to you and plot and plan on how to do that, that means you care too much, and caring too much will dry out a vagina faster then a bag of salt

seriously, go out, and meet women in general, meet alot of them, there is no need to try to only get one at a time, there is absolutely nothing special about any girl, until she proves otherwise, good looks? have you looked around? maybe you don't live in a city or something cause hotties tend to be all over the place, it doesn't make them special or any different then any other girl, they are just people with vaginas

GOOD LUCK

Yeah dude, seriously dont wait too much longer, if you do keep playing it safe she WILL put you in the friendzone

To add onto all of these other reposes, just make sure you know what you're getting yourself into. If she has a boyfriend and decides to hook up with you and keep this guy as well, YOU ARE PRIORITY NUMBER 2. Trust me on that, there is no way I can sugar coat it. You may not be able to get a relationship out of this girl in the first place.