Is empty set element of every set if it is subset of every set?

When $X$ and $Y$ are two sets, we say that $X\subset Y$ if every element of $X$ is contained in $Y$.

With this definition, you see that $\emptyset \subset Y$ for any set $Y$. Indeed, there is no element in $\emptyset$, so every element of $\emptyset$ is contained in $Y$ (trivially true as there is nothing to check).

However, if you want to write $\emptyset \in Y$, this means that there is one element of $Y$ which is a set and that this set is the empty set. When $Y=\{0\}$, you have only one element in $Y$, and this one is not a set, it is a number, which is $0$. Hence, $\emptyset\notin \{0\}$.

Both statements $9a$ and $9b$ are false.

$x \in \{ y \} $ if and only if $x = y$. Thus, $\varnothing \in \{ 0 \}$ if and only if $\varnothing = 0$.

Out of context, there is actually ambiguity here. Often, in set-theoretic contexts, we interpret natural numbers as being the set of all smaller natural numbers; e.g. $3 = \{ 0, 1, 2 \}$. And according to this convention, $0$ is indeed equal to $\varnothing$.

But we might not adopt this convention, and we take $0$ to be its own thing that is unequal to $\varnothing$ or any other set that is 'naturally' written.

There are several ways to represent the empty set. $\{ \}, \emptyset, \text{ and } \varnothing$ are three common ways.

Saying "the empty set(nothing)" is incorrect. The empty set is the set that contains nothing. A bottle can contain nothing, but the bottle itself is something.

Hence, for example, the set $\{\varnothing\}$ is not the empty set simply because it has something in it. In English, the set containing the empty set is not the empty set.

For the empty set to be a member of a set, it has to actually be in that set. The empty set is in $\{1,2,\varnothing\}$. The sets $\{1,2\}$ and $\{1,2,\{\varnothing\}\}$ do not have the empty set in them.

A subset of the set S, is either the set S or the set S with some stuff removed from it.

For example, a subset of $\{a,b\}$ is the set $\{a,b\}$ with $0$ to $2$ things removed from it. These sets are subsets of $\{a,b\}$:

\begin{align} \{a,b\} &- \text{nothing was removed}\\ \{b\} &- \text{a was removed}\\ \{a\} &- \text{b was removed}\\ \{ \} &- \text{a and b were removed} \end{align}

where the last set, $\{ \}$, is the empty set.

Start with any set, take everything out if it, and you are left with an empty set. Hence the empty set is a subset of every set.