I have to equations: P(x)=student x knows calculus, and Q(y)=class y contains a student who knows calculus.
For something like "Some students know calculus" would it be enough to write $\exists x P(x)$, or would a better way be $\exists y \exists x Q(y) \wedge P(x)$?
Similarly, to express "every class has a student in it who knows calculus", would it be sufficient to write $\forall y Q(y)$?
For your first question I think your first expression, that is $\exists x P(x)$ would suffice. Your second candidate is a bit pointless unless you can express the relation between students and classrooms (that is a student belongs to a classroom). If you can express it, then maybe it will be considered a nice addition but I don't think it's required.
For your second question what you wrote is sufficient.
P.S. The command for "for all" is "\forall".