why does "daha cabuk" not work for "faster"
Why does "arabası" not have the -dan ending? Why isn't "Annemin arabasıdan daha hızlı"?
because you are not comparing it to anything, the thing that's compared to gets -den.
e.g. "Annemin arabası senin arabanDAN daha hızlı"
That would mean "It is faster than my mother's car" rather than "My mother's car is faster".