Can I say $\frac{4}{0}\neq$ something? For example,
$\forall x\in\mathbb{R},~\frac{4}{x}\neq0$.
In this quantified statement, $x$ could be $0$ at a moment, leading the $\frac{4}{x}$ to be meaningless. So in this case, is a statement/proposition that includes a meaningless term, such as $\frac{4}{0}$ in $\frac{4}{x}\neq0$ a legal(I mean, it is OK to write so) statement?
The quantifier doesn't change anything, you're still making an undefined statement. It's as invalid as it is in any other context.