Is an "an element of" expression a truth value?

The expression $a\in A$ is a logical statement: it is the assertion that $a$ is an element of $A$. So its "value" is a truth value: either $a$ is an element of $A$ and the statement is true, or $a$ is not an element of $A$ and the statement is false. In most contexts we don't think of logical statements as having "values" though; we just say whether they are true or false (and usually only state them if they are true!).

There are some contexts, however, where the notation $a\in A$ is used with the "value" $a$. For instance, if you say "There exists $a\in A$ such that..." then you are asserting the existence of $a$, not the existence of the truth value of $a\in A$. More specifically, you are asserting the existence of $a$ such that in addition, the statement $a\in A$ is true. Another similar context is in set-builder notation: $$\{a\in A:a^2=a\}$$ refers to a set of values of the variable $a$ (such that $a\in A$ is true and $a^2=a$ is true), not a set of truth values of $a\in A$.

Note that these usages are really just quirks of the grammar of mathematical writing, and are not at all particular to the symbol $\in$. For instance, you can also write "There exists $x>0$ such that..." where you are referring to the existence of $x$ such that $x>0$ is true, not the existence of the truth value of the statement $x>0$.

The notation $a\in A\in B$ is not commonly used, though presumably it would mean "$a\in A$ and $A\in B$". In general, we only "chain together" statements like this when we are talking about a transitive relation. So for instance, we write $x<y<z$ for "$x<y$ and $y<z$", since this also implies $x<z$. But we don't usually write $a\in A\in B$, since this notation misleadingly looks like it would also imply $a\in B$ which is not necessarily the case. (A funny variant on this is that we do commonly chain together $a\in A\subseteq B$, since these relations are "transitive" in that $a\in A$ and $A\subseteq B$ together imply $a\in B$.)

For your first question: The correct option is 3, i.e., it depends on context. For example, you can say

The number $5\in\mathbb{R}$ is a very nice number, and its name in English is "five'.

and you can also say

The statement $\sqrt{2}\in\mathbb{Q}$ is false, as was proven by the Greeks.

For your second question: Yes, that is how it should be read, $a\in A\in B$ means that $a\in A$ and $A\in B$. It's a similar sort of combined notation to saying $x\leq y\leq z$ for numbers, for example.