When we write things like $\forall x$ does it need to be followed by an $\in \mathbb{A}$ for some set $\mathbb{A}$?
Also, sometimes people write things like $\text{for some }x\in(0,3)$:
1)By "$\text{for some}$" do they mean $\exists x $ or $\forall x$?
2) With the interval $(0, 3)$ is it implied that they're considering some set like the reals?
Why? Is it just by context?
A lot of this is contextual.
$\forall x$ by itself implies the set from which $x$ is drawn can be determined from the context (one hopes!)
"For some $x \in (0,3)$ we have P(x)" usually means there exists such an $x$.
$(0,3)$ is taken as an interval of the reals unless there is some other context. E.g., sometimes people will write $[n,m]$ for the integers in that range. However, it should be clear from the context.
(A friend of mine was writing his masters thesis the same time I was and in his work he used $[n,m]$ for a range of integers. Every now and again I saw a page of his from the printer and say, "that result doesn't make sense! ... oh wait, not reals, integers.")