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**Failure** 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, $\displaystyle x\in\mathbb{R}$, and you, the defender of that statement, would have to come up with a $\displaystyle y\in\mathbb{R}$ such that $\displaystyle x=y^2$ 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 $\displaystyle x<0$, otherwise it is quite easily possible to come up with such a $\displaystyle y$

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