Learn Turkish: Kirli araba neden yıkanmıyor? - Why is the dirty car not being washed?

The root verb is yıka- (to wash). Because it ends in a vowel, it takes the passive suffix -n instead of the standard -ıl or -il. This changes it to yıkan- (to be washed). Remember, verbs ending in a vowel or the letter l use a form of -ın for the passive!

Why is the negative part of the verb -mı instead of -ma?

This is because of vowel narrowing. The base negative form is yıkanma-. But in Turkish, when the present continuous suffix -yor is added right after an a or e, it squishes or narrows that vowel into an ı, i, u, or ü. So, yıkanma-yor naturally turns into yıkanmıyor.

Why isn't araba in the accusative case (arabayı) since it is the thing being washed?

Because the sentence is in the passive voice. In an active sentence (like 'They are washing the car'), the car is the direct object and takes the accusative case. But in a passive sentence ('The car is being washed'), the object becomes the subject. Subjects always stay in the nominative (bare) case, so it remains araba.

Could I put neden at the beginning of the sentence?