That's true, may be a slightly confusing way to interpret it, given that C = A∪B. In context I think it's better to read it as P(A|B), in the case that C occurs.

wouldn't that be P((A|B)|C) ?

or even more precedence dubious: P(A|B|C) ?

The "given" symbol should only appear once in any probability statement.

That's true from an objective, mathematical standpoint. But the paradox is really saying something about how humans perceive the statement, so different (mathematically equivalent) ways of framing the same thing can make a difference. It's a minor point.