Learn German: Welches kompakte Auto fährt sich am besten?

This is the reflexive middle voice, which is a common alternative to the passive. We use it to describe a general characteristic of a subject, usually how easily or well something can be done. Saying wird am besten gefahren would focus on the action of a specific driver, whereas fährt sich am besten focuses on the handling qualities of the car itself.

Can I use this reflexive construction to replace the passive with any verb?

No, it only works with certain verbs and is almost always paired with an evaluating adverb like gut, schlecht, leicht, or in this case, am besten. It is mainly used for things being used, handled, or processed. Another common example is Das Buch liest sich gut (The book reads well).

Why does the adjective end in -e (kompakte) instead of -es? Doesn't the neuter noun Auto require an -es ending?

The question word welches functions like a definite article (a der-word) and already carries the strong neuter ending -es. Because the gender and case are already clearly marked by welches, the adjective that follows takes the weak ending -e.

Why use am besten here instead of just das beste?