Does a set include its elements or just its subsets? I am a little confused about the basic definition of inclusion.

I understand that, for example, $\{4\}\subset\{4\}$.

I also understand that $4\in\{4\}$, and that it is false to say that $\{4\}\in\{4\}$.

However, is it possible to say that  $4\subset\{4\}$?
First of all, sets can be elements of other sets too. For example if $X$ is a set then $\mathcal P(X)$ is the power set of $X$, and it is a set whose elements are all sets. But now that's out of the way, let us focus on the question whether or not $4\subseteq\{4\}$ makes sense.

It is possible if you interpret $4$ as a set. In naive set theory we often work under assumptions closer to type theory. There are real numbers, and there are vectors, and functions, and there are sets and there are other sort of type of mathematical objects.

In modern set theory we often work under the assumption that everything is a set. We construct surrogate sets to interpret other concepts such as the integers, or the real numbers, as sets.

For example, one of the mainstream ways to interpret the ordered pair $\langle x,y\rangle$ is by considering the set $\{\{x\},\{x,y\}\}$. Even though ordered pairs are "not sets", we can represent them using sets.

Similarly for integers, we can represent them as sets too. Often we choose the following encoding, $0=\varnothing$ and $n+1=n\cup\{n\}=\{0,\ldots,n\}$. In that case $4=\{0,1,2,3\}$. Clearly under this interpretation $4\nsubseteq\{4\}$. But under this interpretation, $0\subseteq\{0\}=1$.
No, because $4$ is not a set, but an element.