Validity of predicate logic formulas
Not a counter model, for as you say, the antecedent is not true when $x = 0$.
A counter model:  Every number has a number less than it in $\mathbb{R}$, however there are no least number.
A counter model:  For $x$, let $y = 1/x$, so the antecedent holds, there is no $y$ such that for all $x$, we have that $x \cdot y = 2$.
Not a counter model:  The consequent is always true, let $y = \epsilon$, then for all $x$, it holds that $yxy = x$.