Should I ask her out the next time I see her (during a work period for the same project?). OR should I continue to work throughout the semester and try to have fun/flirty conversations and show her my intelligence then ask her out AT THE END.
Don't wait until the end in 9 weeks, you could end up becoming too friendly, and then she'll only think of you as a "friend" and now want to date you. But probably should do it tomorrow either. Give a week or two, at the most, depending on how the interaction goes. Try to strike some casual conversations in class, see if she seem open to engaging you, and if things go relatively well, go ahead and ask her out for the movies or something on a weekend.
What if she rejects me? Will it be super, mega, ultra-awkward?
Not if you don't let it be.
How do you deal with rejection from girls that you will see regularly after they've rejected you?
Not matter how nervous or embarrassed you feel inside, act as if nothing has changed. You're cool, no hard feelings, continue to interact with her as much as is necessary, like you're perfectly fine. That will likely put her at ease, so she won't feel awkward around you when she see that you're cool,, and it will help you legitimately get over the awkwardness you feel
What if you ask out a coworker and get rejected?
Never try to date a coworker.
Or your neighbor, and you still live beside each other for years to come.
Depending on how close you are (are we talking someone who lives in the same apartment building, across the hallway from you? Or just someone who lives in a house down the block from you?), it may be advisable to avoid that too.