Learn Turkish: Hangi seviyede gitar çalabilirsin? - At which level can you play the guitar?

Because you're asking about the general skill of playing the guitar, not a specific, known guitar (like "the guitar"). In Turkish, general, indefinite direct objects usually sit right next to the verb and don't take the accusative case suffix.

How is the verb çalabilirsin built to mean "can you play"?

It starts with the verb root çal- (to play). We add the ability suffix -abil (can), followed by the aorist (simple present) tense suffix -ir for a general fact or ongoing ability, and finally the personal ending -sin for "you".

Why does seviye (level) have the -de ending here?

The suffix -de is the locative case, meaning "in", "on", or "at". Just like in English where we ask "at which level", Turkish uses the locative case to pinpoint the level of the skill.

Does the verb root çal- have other meanings besides playing an instrument?

Yes! While it means "to play" an instrument or a song, it is also the verb for "to steal", "to ring" (like a phone or bell), and "to knock" (like on a door). You can always tell which meaning is intended from the context.