playing A guitar - is wrong, but why? how does it sound to an Eng.speaker?

"Playing a guitar" means playing a single unspecified guitar. So it would be ok to say "Yesterday at the bar, I drank a beer and played a guitar" to suggest that you found a specific guitar there to play. But it would be a bit odd to say "In my spare time, I play a guitar", as that again suggests that there is one guitar that you play. It would be similar to saying "In my spare time, I play my guitar" but vaguer for no clear reason.

The definite article, meanwhile, is used not just to reference a specific instance but to generalise an entire class. So "I play the guitar" essentially means I play any guitar.