In order to be admitted to the college, it's normally necessary to learn and demonstrate an adept level spell, so if learning a magical spell doesn't fit your role-play, you may not want to do the quest line with that character.
If you don't know any magic, but want to learn it at the school, that's definitely an option, the examiner will offer to teach you the spell required for the admittance test for a modest fee if you don't know it, and the only place you can learn the higher level spells is at the school, though you'll need to get your skill up in each school of magic to enable you to learn them.
You can also gain admittance to the school by using your speech skill to convince the examiner that you don't need to perform the test; however, the quest line does involve a few doors that can't be opened without knowing frostbite and flames (though staves might also be an option); the spell tomes required are generally close by, but you'd have to find the staves first if you want to complete the quest line without ever casting a spell yourself.
There are also mastery quests for each school of magic, those quests can only be unlocked by reaching 100 skill in the respective school of magic, so you can't experience everything the school has to offer without actually learning magic, though you should be able to complete the main school quest line.
She may have gotten herself lost during the commotion. Settle your bounty and wait a day or two away from Winterhold, then come back during business hours (just to be safe). If that doesn't help, follow Halfwarr's advice.
Best Answer
From UESP wiki, "First Lessons" quest article:
Talk to Faralda and choose the appropriate conversation option to buy the 'Conjure Flame Atronach' spell. You should now be able to use it.
Also, from the same UESP article linked above: