"Older, wiser, and more mature" person here 😀 You're in luck as I have the time right now and am in a bit of a mood to impart some lengthy advice. I'm mostly responding to what your first post says, even though it may appear to jump around a bit.
The "Just do it" advice is good, but you're looking for a little more reassurance, and I can understand that 😀
"Don't do it, she's American" is rubbish, and is coming from the unexperienced.
All your "Reasons to ask her to coffee" are good reasons!
All your "Reasons to forget about it" is just the usual scared inner voice stuff that comes up in everyone when they are otherwise compelled to do something like this. Appreciate that this is a biological mechanism that's designed to protect you, but it goes into overdrive and over compensates. In nature, when it could mean the difference between being eaten by a lion or not, being safe is better than being sorry. However, in these situations, living with the regret of never doing anything and wondering what could have been is worse than the repercussions, 99.9999999 percent of the time. What I'm saying is it's one of those things in life that you just do and if you don't do it because of the small chance of what could happen badly, then you'd never enter a vehicle because you might be in a car crash one day, etc.
Here's some food for thought;
If every guy avoided asking a girl out because;
a. She MIGHT be in a relationship, or...
b. She MIGHT be being asked out by other guys, or...
c. Some other guys MIGHT even just be interested in her...
...no one would procreate and we would have died out long ago, plain and simple.
The fact is, women are usually getting a lot of attention, and this is normal. So it just becomes a case of - of all the guys that are interested in her, are you the one that is right for her? (And is some cases - simply just the one that has the guts to approach her?)
And you have to assume the answer is "YES" until circumstances dicate otherwise, otherwise you'll be single forever.
To me it sounds like a great set of common interests that you have which mean, yes you should do something about this. This is worth finding out.
And here's the trick - all the negative outcomes that you've thought of (and a couple people have even mentioned, like awkwardness if you have to see each other again) this all comes down to how you do it, and how you react afterwards. Yes, it all depends on that.
Yes that girl you went to coffee with could have been up front about already having a boyfriend, but perhaps you could have been MORE up front about your intentions too. She either saw you as a friend, or perhaps thought you seemed lonely and wanted to boost your ego and do something nice. Regardless of what you think about that, there's many reasons why she might do that, all which can be made simpler by the way you do what you do.
When I ask a girl out to coffee, because of the way I do it, if she's taken, she'll be flattered (important to note that I get this result), and then kindly let me know she's taken, but thank me. Eg. the last time it happened, we'd had a nice pre-amble chat, and then I said "Well, this was fun - would you like to go out for coffee tomorrow?" and it's my tone of voice and body language which make this a positive experience for both of us instead of awkward, and she replied light-heartedly "Oh - thank you for the offer, but I have a boyfriend currently and he might not be ok with that!" I said "Ah, of course, someone as attractive as you already has a boyfriend" in my overly theatric way, and we had a little laugh, then a little more small talk - and when we parted you could tell I'd made her day. I bet she told her friend a guy asked her out today.
I know I sound extroverted from that, but that's just my style. Introverts have their own style too, don't worry. They can be really friendly and calming to talk to. And you have your particular style, maybe you haven't discovered it yet. The main takeaway from this is, both of us can ask the same question and it can be a positive experience, not an awkward one, all because of the way we did it.
So what is the right way? The one where you appear comfortable doing what you're doing, and considerate and respectful of her as a person. (Or you could be the kind of guy that gets away with naughty jokes etc. and that's cool - horses for courses, and time and place make all kinds of things possible.)
First get rid of the idea that she would be "offended" that someone finds her interesting and even attractive. That is ridiculous. At worst, it would be awkward - because of the way you did it. And that is totally avoidable too, as I've been saying.
You feel "sick to your stomach" about it, why? Because you really really want to do it. But at the same time you're scared of putting yourself out there, like the effect public speaking has on people. So it causese a wierd conflict! You are really compelled to pursue this - but an annoying inner voice is popping up and stopping you.
Anyway - how to appear comfortable. You don't have to actually be comfortable, you just have to appear comfortable. Yep. Not only does this work for many people in many social situations, including myself (even when I was more stressed in my younger years) but I also have performance experience interacting with people etc. live shows, as part of part time work that I did for a few years, and the same rule applies there - if you look confident or comfortable, then they beleive you are. Even though inside you might be going "omg omg omg omg", hahah. It's true, and it works. You'd be surprised - that's how a lot of cool things get done.
Channel the nervous energy into something productive. Use it to motivate yourself and be aware of your appearance and how you sound. Take your time to think about what you'll say next - you have a lot more time than you realise!
Above all - be direct. Women really appreciate you having the guts to be up front, just as much as you wished the coffee girl was up front about having a boyfriend.
Depending on the situation, such as yours (you're in a group project, and it could seem wierd for the rest of the project) you can do some sort of "disclaimer", but don't make yourself sound small. Eg. you can start with "I don't usually do this, but...", but don't start with "Hey this is only if you want to..." that sort of thing. The difference between those two statements and the way you deliver them is huge. "I don't usually do this, but..." sets the tone that she's THAT interesting that you're willing to go out on a limb to do this. "Hey only if you want to..." kind of a statement sets the tone that you're expecting to be rejected, which turns women off.
Use your own judgement of when or where would be a good place to bring it up, or how to talk to her in private, but don’t put it off forever!
If she politely declines, it's super important you give a very confident response and say something like "Ok no worries - I had to ask though!" or "Well it was definitely worth a shot!" that sort of an answer, and then move on with the project like nothing happened, that will actually win you big social points as a guy who has guts but is also a gentleman and confident with himself.
Ok I've said a whole bunch and you'll probably get information overload soon. So go away, have a think about it, visualise it going well like a conversation between two adults, and if you like you can post back here with your thoughts if you're still unsure, or just go and try it. With women it's all about "seize the day" and striking while the iron's hot, doing it while you're in the mood and have the motivation to say what you mean and not over think things.
Don't worry - I've been in friend circles where we know (secretly) that one guy asked another girl out, and she said no, but it's all cool and we all still hang out without that being a thing. We're men, it's our job to put ourselves on the line and do this. People get that.
Oh yeah - if you're nervous about speaking in general, then warm up - spend the earlier part of the day, or week even, just chatting to random people, try to make the conversation light hearted, say something positive, etc.