In the very obvious cases where x == y, or x == not y, or x == y + 1, or many other correlations.

Only if y is known to be dependent on x. If y is random but just so happens to be `not x` or `== x` then that won't make the result any less random.

0 is as random a number as 529890873740477 is.

Single numbers don't really have entropy at all, so that is not something it even makes sense to talk about.