Learn Turkish: Benim arabam senin araban kadar hızlı.

In Turkish, to say "as [adjective] as", you put kadar right after the thing you are comparing to, and then add the adjective at the end. So, senin araban kadar hızlı literally translates to "your car as-much-as fast".

Could I drop the words benim and senin and just say arabam araban kadar hızlı?

Yes, you absolutely can! The possessive suffixes -m in arabam (my car) and -n in araban (your car) already tell us who the cars belong to. Using benim and senin here just adds extra emphasis, like saying "MY car is as fast as YOUR car."

Does the word before kadar need a special case ending?

Not when it is a regular noun or a possessive phrase like senin araban (your car). However, remember that if you are using kadar directly with standalone personal pronouns, they must take the genitive (possessive) ending, like benim kadar (as much as me) or senin kadar (as much as you).

What happens if I want to make this sentence negative (My car is NOT as fast as your car)?

You simply add the negative word değil (not) at the very end of the sentence. It becomes: Benim arabam senin araban kadar hızlı değil.