In the following logical expression, is it incorrect to share the variable $x$ symbol with the universal character and the inequality?
$$\forall x{>}0 \left( \frac1x >0 \right)$$
Also, is the conditional in this expression correct?
$$∀x(x > 0) → \frac1x > 0$$
$x>y$, or any binary relation $R$ on some set $S$, for that matter, is just $R \subseteq S\times S$. In other words, $x>0$ is just $R(x,0)$ which is a particular instance of the following $R(x,c)$ where $c$ is some structural constant.
What you are writing is basically, $\forall x \in S$ if $R(x,c)$, then ... $F(x)$. In summary, they are both correct. That is, you can say for every $x$ greater than $0$, some formula. Or, you can say, for every $x$ in $S$, if $x$ is greater than $0$, then some formula.
$$\forall x \in S, x>0, F(x)$$ is contexually that same as $$\forall x >0, F(x)$$ (domain is implicit here, although i wouldn't suggest you omit domains like you have done in order to avoid ambiguity)
which is contextually the same as $$ \forall x \in S, x>0 \implies F(x)$$