Learn Turkish: Niçin okula gitmiyorsun? - Why are you not going to school?

Yes, absolutely! Niçin and neden both mean "why" and are usually interchangeable. Neden is a bit more common in everyday spoken Turkish, but niçin (which originally comes from ne için, meaning "for what") is also perfectly correct and natural to use here.

Why does okul have an -a at the end?

The -a is the "to" suffix (often called the dative case). Because the action is directed to or toward the school, okul (school) becomes okula (to school).

The root verb is gitmek (to go). Why doesn't the t change to a d here, like it does in gidiyorsun?

The t in gitmek only changes to a d when it is followed directly by a vowel. In this sentence, the next letter is m from the negative suffix -mi. Since m is a consonant, the t does not change and stays exactly as it is: gitmiyorsun.

The negative suffix is normally -me (as in gitme, meaning "don't go"). Why is it -mi in gitmiyorsun?