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 $\;\;3 \nsubseteq \{1, 2, 3, 4, 5\}, \text{... but}\; 3 \in \{1, 2, 3, 4, 5\}$.
The notation $a\in B$ denotes that the element $a$ is actually a member of the set $B$.
The notation $A\subseteq B$ denotes that every element of $A$ is also an element of $B$.
In your example: every element in the empty set $\{\}$ is certainly also an element of $\{x\}$... this is vacuous, because there ARE no elements in $\{\}$.
On the other hand, $\{\}$ is not an element of $\{x\}$; if it were, your set would look like, for instance, $\{x,\{\}\}$. It would literally mean that the empty set was an element of the set in question... but in this case, that set contains only one element, namely $x$.
Best Answer
If A is a subset of B this means all elements of A are elements of B. If A is a proper subset of B this means that all elements of A are elements of B but there is at least 1 elements of B which is not an elements of A.
In the notation you are using a subset is shown by ⊆ and a proper subset by ⊂.
If you check how these definitions apply to your questions
Every element of $\emptyset$ is an element of $ \emptyset$ because there are none - there is no element of $\emptyset $ which is not an element of $\emptyset$. This shows that (a) is false while (c) is true.
The empty set is in fact a subset of every set (not necessarily a proper subset) - there is no element in the empty set which is not in another set whatever the other set. So (b) is true. (d) is also true because the set {$\emptyset$} contains an element, $\emptyset$, whereas $\emptyset $ contains no elements.
Take note of previous comments on notation: it is more usual to notate a subset as $\subset$ and a proper subset as $\subsetneqq$.