You are correct for the first three, but not the fourth. To say that
$$A \subseteq B$$
means that every element of $A$ is also an element of $B$. The only element of $\{x\}$ is $x$ itself, and the only element of $B$ is $\{x\}$. These aren't the same, so the statement
$$\{x\} \subseteq \{\{x\}\}$$ is false.
In short, $x \notin \{\{x\}\}$, since the only thing in $\{\{x\}\}$ is $\{x\}$.
Subset is more general than proper subset (like green is more general than dark green -- everything that is dark green is green, but not everything that is green is dark green).
A proper subset of $A$ is a subset of $A$ that is not equal to $A$. So if $A = \{1, 2\}$, then the subsets of $A$ are $\emptyset$, $\{1\}$, $\{2\}$, and $\{1,2\}$.
The first three are proper subsets of $A$ since they are subsets of $A$, but they aren't equal to $A$. The other subset of $A$, $\{1,2\}$, is not a proper subset of $A$, since it equals $A$.
So basically, for any non-empty set $A$, if you think of all of its subsets, then all but one of them is proper. The only one that isn't proper is $A$ itself.
Note, if $A = \emptyset$, then its only subset is itself, so that shows you the empty set has no proper subsets, since it only has one subset, which is $\emptyset$.
Best Answer
You are correct in saying that $$ \emptyset \subset \{7,8\}$$
because every element of $\emptyset $ is an element of $\{7,8\}$ but$\{7,8\}$ has two elements which are not element of $\emptyset $
On the other hand $$ \emptyset \subseteq \{7,8\}$$ because every element of $\emptyset $ is also an element of $\{7,8\}$
Of course we understand that $\emptyset $ does not have any element so it is a subset of every set.
Note that $\emptyset $ is not a proper subset of itself, but it is a subset of itself.