$\exists x \in A \left[\varphi(x) \right] \rightarrow \exists x \varphi(x)$ and $\forall y \psi(y) \rightarrow \forall y \in B \left[\psi(y) \right]$ Here are two conventions that I have been told to use:
\begin{align}
&1)\  \exists x \in A \left[ \varphi(x)\right] \text{ should be interpreted as } \exists x \left [x \in A \ \land \ \varphi(x) \right]\\
&2)\  \forall y \in B \left[ \psi(y)\right] \text{ should be interpreted as } \forall y \left [y \in B \rightarrow \psi(y) \right]\\
\end{align}
Now, when I skim logic posts on this site, I will frequently see examples that use expressions such as $\exists x \varphi (x)$ or $\forall y \psi(y)$. I am not sure how to interpret these in the context of the aforementioned convention.
If I see something like $\exists x \varphi(x)$, should I assume that $\varphi$ must be some conjunctive proposition where the first conjunct is of the form $x \in D$? i.e. $\varphi(x):= x \in D \land \alpha(x)$ . Similarly, if I see $\forall y \psi (y)$, should I assume that $\psi(y):= y \in E \rightarrow \beta(y)$?
My thinking is no, that is incorrect. Given this, it seems as though the following definitions of $\varphi$ and $\psi$, and the way that they are subsequently used with quantifiers, are well defined formulas, but as to whether or not they evaluate to true/false is a different story:
Let $\varphi(x):= x\in A \rightarrow \alpha(x)$. Then $\exists x \varphi(x)$ expanded is $\exists x \left[ x \in A \rightarrow \alpha(x) \right]$.  Depending on whether or not the system we are in allows for a universal set, this could be false: for example, if $A$ is the universe and no element in $A$ has the property $\alpha$. If the system does not allow universal sets, this is always true.
Let $\psi(y):= y\in B \land \beta(x)$. Then $\forall y \psi(y)$ expanded is $\forall y \left [ y \in B \land \beta (y) \right ]$. Depending on whether or not the system we are in allows for universal sets, I suppose this statement could be true: for example, if $B$ is the universe and every element has the property $\beta$. However, if our system does not allow universal sets, this is always false.

If everything I have said is correct, I would like to turn my attention to the major point of this post.
Are the following two statements true regardless of whether or not a universal set is allowed:
\begin{align}
&1) \ \exists x \in A \left[\varphi(x) \right] \rightarrow \exists x \varphi(x)\\
&2) \ \forall y \psi(y) \rightarrow \forall y \in B \left[\psi(y) \right] \\\
\end{align}
I think the answers to both are yes. I am mostly interested in the first one because I find that, perhaps in attempt to be informal, many people will write $\exists x \varphi(x)$ when it is more practical to write, for example, $\exists x \in \mathbb R \varphi(x)$.
 A: If you are working in set theory where the underlying logic is 1-sorted first-order logic, then you are correct that the quantifiers $\exists x \in A$ and $\forall x \in A$ should be viewed as shorthand. This is not the only way to approach set theory, and arguably it is not a natural way to approach set theory. But let's leave that discussion behind and talk about how people use quantifiers in practice.
In practice, people tend to implicitly use bounded quantifiers when they use an unbounded quantifier.
For example, let's say I write something like
$$ \forall n \exists m (n = 2m \lor n = 2m + 1)$$
Hopefully, it's clear from context that $n$ and $m$ are variables which range over integers (or natural numbers, depending on context). Clearly, it doesn't make too much sense to say that $\mathbb{R} = 2m \lor \mathbb{R} = 2m + 1$, for example, no matter what $m$ is. So it wouldn't be natural for us to actually have a totally unbounded quantifier on $n$.
The most common convention mathematicians use is actually aligned with the conventions of many-sorted logic. The idea is that each variable name implicitly has a "kind". All quantifiers over this variable name are implicitly only quantifying over things of this kind.
Generally, you'll see conventions like

*

*$n, m$ are integers

*$x, y$ are real numbers

*$z, w$ are complex numbers

*$\alpha, \beta$ are ordinals

*$f, g$ are functions (usually between two sets which are known from context)

*$U, V$ are open sets

*$A, B$ are sets

*$\mathcal{A}, \mathcal{B}$ are families of sets

et cetera.
You will then know that whenever you see a statement of the form $\forall n P(n)$, you should interpret this statement rigorously as $\forall n \in \mathbb{N} P(n)$ (which would then be interpreted formally as $\forall n (n \in \mathbb{N} \to P(n))$).
And you will know that whenever you see a statement of the form $\exists \alpha P(\alpha)$, you will have to interpret this statement as $\exists \alpha (isOrdinal(\alpha) \land P(\alpha))$.
So the key here is that when you see a statement of the form $\exists v \psi(v)$, you should actually reinterpret the quantifier and not reinterpret $\psi$.
