I ended the game but still don't know what's the purpose of vinyl record that I gained from robot in front of courthouse. Is it playable somehow?

Well, the only record player in the entire game seems to be the one back at the ship, and that one isn't accessible.

So i am wondering about this as well (plus a couple other things). >_>

It's just there for flavor, and to reinforce Horatio not knowing what humans are.

We got that primarily from him finding a "ancient robot" at the dome, i'd say. The record shows up at a point where we're long past that. :)
(...wait, he does have a record player. Didn't he use it on his ship, y'know, with a record?)

Meanwhile: That musical robot? I liked that one. Got a nice tune to her.

Oh, he knows what a record is. Just not what children are. I agree that the skeleton does most of the work. Although it's interesting to imagine his reaction when he finally listens to the record back at the ship! I also thought -- though perhaps this is cheap sentiment -- that there might be some pathos in having a relic of *children* as opposed to just humans in general.