あの女の子は天才だ
あの女の子は天才い
あの女の子は天才です
In all of them I take the meaning to be, “That girl is a genius.”
However, the endings are different. From what I understand, the first two describe the girl as having genius qualities, whereas the third is that the girl is a genius.
What is the difference in the ending between the first two, and are all of the three above usages correct?
I’m no grammar expert, and I can’t answer you for the second sentence, but: だ, as it is used in the first sentence, is an auxilary verb associated with the plain form that states that is used to say that something “is” (Like “That girl is a genius”), while です, as it is used in the third sentence, is considered to be the polite version of it, but as far as I know, it’s technically not. です is simply a politeness marker (you can read more about it here for だ and here for です).
Again, I’m not a grammar expert, and I can’t give you an answer for the second sentence, but I hope this helps you at least a bit 
For simplicity sake you can consider the first and third the same, but the first is in “plain form” and the third in “polite form”. You might use the former when talking to your friends and the latter when talking to someone older than you, a teacher, your boss, etc. @NeoArcturus is correct that technically です isn’t just a polite version of だ, but that’s best left until you’re further along in your grammar studies.
As for the second, that is not grammatical. You may be getting confused with い-adjectives, which all end in い. You can’t simply append い anywhere you want.
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