I've seen it dozens of time. It is like a simple perfect little story. I wish the pool technically was just a little better but regardless it really is great. I read a book on the Twilight Zone and it said in the original story the Klugman character loses the game. When he is waiting to die, Fats Brown says something like, "Don't worry, your not going die, at least not in the way you thought". Oh yes, you will die in time like everyone and be forgotten like all those that are second rate, but champions live forever".
Just a guess, but the cues look like Martin cues. I would guess this was filmed in Southern CA. and at the time he would have been the cue makers of note around there. I could see the technical advisor supplying the cues and Martins would have been the logical choice. Back then, 1961, there wasn't really that many cuemakers period.