Why $x$ has superscript $i$ in the expression $\partial/\partial x^i$ (Differential Geometry)? Let's start with an example. Let $X,Y$ be smooth vector fields on a smooth manifold $M$, then the Lie bracket
$$
[X,Y]=\left(X^i \frac{\partial Y^j}{\partial x^i} - Y^i\frac{\partial X^j}{\partial x^i}\right) \frac{\partial}{\partial x^j},
$$
where the summation convention applies.
I understand why $X^i$ and $Y^i$ has superscripts - they are components of tangent vectors - this is just a simple rule of tensor notation. Components of elements of the cotangent bundle $T^*M$ should have subscripts instead.
However, why does $x^j$ have superscript? $x^j$ is NOT components of a tangent vector or covector. It is just the components of a point in $\mathbb R^n$, which is indeed a function of $p\in M$ ($x^j=\phi(p)\cdot \mathbf e_j$, where $(U,\phi)$ is a chart). so why are we using a superscript rather than a subscript?
Of course, in relativity, we use $x^j$ for the components of a (contravariant) vector, but I need more explanation.
How are the super/sub-script rules of $x$ and $X$ related?
 A: As mentioned in the comments, $X^i, Y^i$ etc are not vectors (on a similar note, something like $\omega_i, \alpha_i$ etc are not covectors). These are components of a vector with respect to a certain basis. Usually, $(x^1, \dots, x^n)$ are used to mean the coordinate functions of a certain chart (or by abuse of notation, the image of a point in the manifold under the chart map).
So, elements of the tangent bundle $TM$ are called (tangent) vectors to the manifold $M$, and denoted by a single symbol, say $X,Y,\xi,v$ whatever else comes to mind. Elements of the cotangent bundle $T^*M$ (which are called covectors) are also denoted by single letters, say $\alpha, \beta, \omega, \eta, \theta$.
Once we have a chart $(U,\phi)$, with coordinate functions labelled as $x^i := \text{pr}^i_{\Bbb{R}^n} \circ \phi$, we can talk about the chart induced tangent vector fields $\dfrac{\partial}{\partial x^i}$, and also the chart induced covector fields (1-forms) $dx^i$. With this, we can take a vector $X \in T_pM$, or a covector $\omega\in T_p^*M$, and expand them relative to these bases as
\begin{align}
X= X^i \dfrac{\partial}{\partial x^i}(p) \quad \text{and} \quad \omega = \omega_i\, dx^i(p)
\end{align}
for uniquely determined numbers $X^i, \omega_i\in\Bbb{R}$ ($1\leq i \leq n$).

Why do $x^i$ come with upstairs indices even though they're merely coordinate functions, and not the components of a tangent vector to the manifold? The way I look at it is that you just need some way of writing things down, and from what I understand, they were originally all written as $x_1, \dots, x_n$ (see the comments in Spivak's Differential Geometry Volume 1, Chapter 4 on tensors, in particular page 114-115). Apparently historically, a covariant vector was defined along the lines of "a thing with indices which transformed like the coordinates $x_i$", which suggested the use of $\omega_i$ as the notation for (components of) covectors. But then in order to make things work out with the Einstein summation convention, the indices were pushed up to $x^i$.
I probably didn't explain this last part very well, but if you have access to Spivak's book, I highly recomment you take a look at it; he does a very good job contrasting and bridging the gap between the classical language and the modern language.
But honestly, the actual math is completely unaffected by your placement of indices. As long as you know what the numbers are and what the actual vectors/covectors are, writing things like

let $(x_1, \dots, x_n)$ be a local coordinate system and consider the covector
\begin{align}
\omega &= \sum_{i=1}^n \omega_i \, dx_i(p)
\end{align}
(with the summation sign explicitly written out)

is perfectly fine from a logical standpoint (though I'm not saying you should upset the traditional notational conventions, just because you can). Of course, it is our desire to keep the summation convention of "one up one down" which motivates a certain choice of index placement.
