So, say you're given ∃x P(x) and ∀x Q(x)-->˜P(x). I want to use this given statements in a formal proof. To manipulate them in the way I want to, I believe I need to use instantiation to remove the quantifiers. However, I am not sure how that works with these separate quantifiers (using the same variable and domain).
If I turn the first statement into P(c), can I also turn the second statement into (Q(c)-->˜P(c))?
Yes, universal instantiation gives you (with minor inconsquential variations in phrasing)
$$ (\forall x \; \phi) \to \phi[x\mapsto t] $$ for any term $t$ whatsoever of your own choosing (as long as $[x\mapsto t]$ denotes proper capture-avoiding substitution).
You're definitely allowed to let $t$ be a term that already has a role in your proof -- such as one you got from instantiating $\exists x\, P(x)$ earlier in the proof.