It's not 0, it's "an unknown positive number" drawn arbitrarily from a set whose lower bound is 0. It's practically 0, but it's some fixed positive number for every instance of the game.

It's equivalent to this much simpler game: "I am going to give you a positive real-number amount of dollars." No matter what I do, you will win money playing this game, guaranteed! Now, how much would you pay to play this game? $0, because for any amount you'd pay to play, I could arbitrarily choose to give you less in winnings.

Like most problems involving infinity, it's unintuitive/paradoxical because it pretends to model a physically plausible scenario but actually doesn't.

>> But for any two finite numbers, A and B, the chance that C will fall between them if selected from an infinite range is 1/infinity (or 0).

I think you're assuming we pick C uniformly at random from an infinite range, but this is not possible. In general, the claim is not true: consider picking C from a sample of a standard normal, if A = -B = 2 then with >95% C will lie between A and B.