Right, on both counts.

Right.

Wrong! The point is not that there must exist a single y that does it for all x. The idea is that if the statement were true, then one could provide an arbitrary, but fixed,

, and you, the defender of that statement, would have to come up with a

such that

holds. If someone comes up with a different x, you are allowed to choose a different y. This is because the existential quantifier for y occurs in the scope of the quantifier of x. So at the point where the y must be show to exist, the value of x has already been decided upon and cannot be changed anymore, so to speak.

This is impossible only if

, otherwise it is quite easily possible to come up with such a

Correct, and this statement is, of course, true.