Both are correct. I can't think of any difference in meaning at all. I can't explain why an English speaker would choose one or the other.

They are both correct. Adding "the" seems more polite.

Normally, they mean the same thing. However, to specify a specific guitar you should use "the". For example, suppose there are two musical instruments in the room, a guitar and a mandolin. Somebody asks you which one you would like to play. It might be better to say "I would like to play THE guitar" to indicate your preference for that specific guitar.

as "guitar" is countable it should be "I would like to play the guitar."