"The girl is not learning."
Translation:Das Mädchen lernt nicht.
No, that word order isn't possible in German. See here for more information:http://german.about.com/od/grammar/a/The-Position-Of-Nicht.htm
The verb ("lernt") should always be the second element in the sentence, which would be right after "das Maedchen".
'Mädel' is rather colloquial. It also carries a somewhat ironic undertone, so it's tricky for non-natives to know when it's appropriate and when not. I suggest to just stick to 'Mädchen'.
Would I be able to say Die Madchen because we're talking about a girl?
No, German girls are neuter -- grammatically, I mean!