Can you Use $x \in S$ with Interval Notation? If I have some variable $x$ and I want to show that it takes values from the set of integers $\{1, 2, 3, \dots, n\}$, is it correct to write the following:
$$x \in [1, n]$$

The default notation is that $[1,n]$ is an interval in the set of real numbers:
$$[1,n] = \{x \in \mathbb R \mid 1 \le x \le n\}
$$
and this includes non-integer numbers such as $x = 1.5$ (assuming $n \ge 2$). So no, that's not correct (unless you explained very carefully and very clearly, in what you are writing, what your intention is for the notation $[1,n]$).
What would be correct is to use the intersection operator $\cap$ and to write $x \in [1,n] \cap \mathbb N$, using the standard notation $\mathbb N$ for the set of natural numbers (a.k.a. the positive integers).
However, it is common notation to literally use the ellipsis in this situation,
$$\{1,...,n\} = [1,n] \cap \mathbb N
$$
If you used that in your mathematical writing, you would be almost universally understood.
