Is A a proper set of its power set? I originally believed that the answer is yes, as all elements of A are subsets of A that would be included in its power set. And, the empty set would be in the power set but not in A.
However, the textbook states that this is sometimes but more often not the case. Would someone mind explaining?
 A: No.  There is a world of difference between a thing, and a set containing that thing.  It is the same as the difference between a cat and the word cat.
Take a set of children: A = {Sam, Claire}.  Its power set is {{Sam, Claire}, {Sam}, {Claire}, {}}.  Note that every element of the power set of A is a set.  Every element of A is a child.  Therefore A is not a subset of its power set, and not being a subset is not a proper set either.
A: Sets that are subsets of their power sets are quite special. They are called transitive sets. If you work through the definitions you will find that $A \subseteq \Bbb{P}(A)$ means that for any $x \in A$ and for any $y \in x$, $y \in A$, i.e., it says that members of members of $A$ are members of $A$ (hence the name transitive sets, because the membership relation on the members of $A$ and their members is transitive). If (as is usual in set theory) you use the von Neumann representation of the natural numbers:
$$
\begin{align*}
0 &= \emptyset \\
1 &= \{0\} \\
2 &= \{0, 1\}\\
&\ldots
\end{align*}
$$
then the natural numbers are all represented by transitive sets while
$\{0, 2\}$ gives you the simplest example of a set that is not transitive (because $2 \in \{0, 2\}$ but $2 = \{0, 1\} \not\subseteq \{0, 2\}$).
A: Supppose $\;\;\mathcal{P}(A)\;\;\;$is the power set of $\;\;A\;.\;\;$
Generally,  the elements of $\;\;A\;\;$ and that of $\;\;\mathcal{P}(A)\;\;\;$ are not comparable and they are of different types. For example, if $\;\;A=\{\;1\;,\;2\;\}\;\;\;$ then  $\;\;\mathcal{P}(A)\;=\;\{\;\Phi\;,\;\{1\}\;,\;\{2\}\;,\;\{1,2\}\;\}\;\;\;$and in this case $\;A\;\not \subset\;\mathcal{P}(A)\;.\;\;$ 
But, we can easily construct an example of class in which  an element can also be considered as a subset of the power set. Example, $\;A\;=\{\Phi\;,\;\{\Phi\}\;,\;\{\Phi\;,\;\{\Phi\}\}\;,...\}\;$
The indistinguishability of elements and subsets of classes leads to logical inconsistency, in general discussions, such classes are avoided by (restricting the class  ) certain axioms like no set is an element of itself, no element of a set is subset of  that set etc.
The use of null set is also confusing for the beginners,as it is of different nature in different contexts, regarding the nature of its elements (though it has not a single element!) 
