The empty set is indeed a set (the set of no elements) and it is a subset of every set, including itself. $$\forall A: \emptyset \subseteq A,\;\text{ including if}\;\; A =\emptyset: \;\emptyset \subseteq \emptyset$$
$$\text{BUT:}\quad\emptyset \notin \emptyset \;\text{ (since the empty set, by definition, has no elements!)}$$
That is, being a subset of a set is NOT the same as being an element of a set: $$\quad\subseteq\;\, \neq \;\,\in: \;\; (\emptyset \subseteq \emptyset), \;\;(\emptyset \notin \emptyset).$$
$\emptyset \;\subseteq \;\{1, 2, 3, 4, 5\},\quad$ whereas $\;\;\emptyset \;\notin \;\{1, 2, 3, 4, 5\},\;$.
$\{3\} \subseteq \{1, 2, 3, 4, 5\},\quad$ whereas $\;\;3 \nsubseteq \{1, 2, 3, 4, 5\}, \text{... but}\; 3 \in \{1, 2, 3, 4, 5\}$.
You have a few mistakes:
$\emptyset \subseteq A$ is $\textbf{true}$ as every single element of $\emptyset$ (there are none) are included in $A$.
$\emptyset \subset A$ is $\textbf{true}$ for the same reasoning as above and because $\emptyset \neq A$
$\emptyset \subseteq \{\emptyset\}$ is $\textbf{true}$ for above reasons
$\emptyset \subseteq \mathcal{P}(A)$ for above reasons
$\{\emptyset\} \subseteq \mathcal{P}(A)$ is $\textbf{true}$ because every element of $\{\emptyset\}$ is contained in $\mathcal{P}(A)$ as $\emptyset \in \mathcal{P}(A)$.
Remember that the power set is a set of sets, i.e if $C = \{a,b,c\}$ then
$\mathcal{P}(C) = \{\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$
It seems like you have misunderstood the subset relation $\subseteq$. We write $A \subseteq B$ whenever every element of $A$ is included in $B$. If $A$ has no elements, this is trivially true.
Best Answer
"is an empty set an element of {empty set}?"
Yes, the set {empty set} is a set with a single element. The single element is the empty set. {empty set} is NOT the same thing as the empty set.
" is an empty set a subset of..." STOP!!! The empty set is a subset of EVERY set. (Because the empty set has no elements so all zero of its elements are in every other set. Or if you take A and B, A $\subset$ B means A doesn't have any elements not in B. The element doesn't have any elements not in B so empty set $\subset B and it doesn't matter what B is.
"is an empty set a proper subset of ..." Yes. A proper subset is a subset that isn't the same set. empty set is not {empty set} so it is a proper subset.