$f$ is said to be strictly increasing $\iff \forall x,y \in D_f:x<y \implies f(x)<f(y)$
I haven't taken any mathematical logic classes yet so I don't know if it correct to write this as well:
$f$ is said to be strictly increasing $\iff$ $$ \forall x,y \in D_f:x \ne y, f(x)<f(y) \implies x<y$$
Is this correct? Would it make a difference if I wrote the $\forall$ at the end, instead of at the beginning?
On
There's no need to specify $x \ne y$ as one of the hypotheses since that is implied by the fact that $x<y$ (or that $f(x)<f(y)$). However, one mistake you made is you wrote the converse of the implication, which is not equivalent to the original implication. Indeed, it's possible for a strictly increasing function to make the statement $\forall x,y \in D_f : f(x)<f(y) \implies x<y$ false, namely any function whose domain has one element.
In formal logical notation, all quantifiers must appear before the things they quantify, meaning that your $\forall x,y \in D_f$ must appear at the front.
That said, there is a widely understood convention in informal mathematical notation that front-loaded universal quantifiers can appear after a statement, in line with the way we often speak mathematics in English (e.g. "$x^2$ is nonnegative for all real values of $x$"). This is best avoided if you're trying to be formal, because you run into ambiguity when instances of quantifier alternation occur.
Suppose $f$ is a constant function. It is then not strictly increasing. But your quantified conditional is true as the antecedent is always false.