Subsets and Proper Subsets equality If $A$ is a subset (not proper subset) of $B$, does that mean that $B$ is a subset (not proper subset) of $A$ and that $A=B$?
 A: Your parentheses give me cause for concern as they introduce a degree of possible ambiguity.
The following is true:
$$(A \subseteq B) \wedge (A \not \subset B) \Rightarrow A=B.$$
Read this as "If $A$ is a subset of $B$ and $A$ is not a proper subset of $B$, then $A=B$".
However, the following is also true: 
$$(A \subseteq B) \not\Rightarrow (B \subseteq A).$$
Read this as "If $A$ is a subset of $B$ then it does not necessarily imply that $B$ is a subset of $A$".
If you were to use the intended notation ($\subset$ for "proper subset" or $\subseteq$ for "not proper subset" a.k.a. "improper subset" or just "subset"), or you use the statement "$A$ is an improper subset of $B$", or you were to introduce the statement "$A$ is a subset of $B$ and $A$ is not a proper subset of $B$", then there would be no ambiguity.
A subset ($\subseteq$) of another set may or may not contain every element of the other set (and it contains no elements that are not in the other set). A proper subset ($\subset$) of another set contains some, but definitely does not contain all, elements of the other set (and it contains no elements that are not in the other set).
A: Yes. We have $A \subseteq B$ but $A \not\subset B$. Of the second statement $A \not\subseteq B$ or $A= B$ (this is the negation of $A \subset B$ equivalent to $A \subseteq B$ and $A \ne B$), therefore $A=B$. It follows that $B\subseteq A$ too.
