I am in doubt if the following is considered a rational expression:
$$\dfrac{1}{x^{-2}}$$
From what I understand, rational expressions cannot have a negative exponent. However, is it okay to have a negative exponent in the denominator?
Also, can $x$ alone be considered a rational expression?
It is subtle.
A rational expression is of the form
$$\frac{a_0+a_1x+\cdots +a_nx^n}{b_0+b_1x+\cdots +b_mx^m}.$$
You just replace the $a_i$ and $b_i$ by actual numbers, then what you obtain is a term which is considered to be a rational expression. In this form, terms like
$$1/x^{-2}\qquad\text{or}\qquad x$$
are not rational expressions, because they are not of the form above.
Then there is something like a rational function, which is a function $f(x)$ that can be computed by a rational expression as above. Take for example the function $f(x)=1/x^{-2}$. While $1/x^{-2}$ is not a rational expression, we can transform it (using equivalence transformations) to such one:
$$1/x^{-2} = x^2 = x^2/1.$$
So (for a second, ignoring the case $x=0$), we can also write $f(x)=x^2/1$, where the term on the right is evidently a rational expression. So the function given by $1/x^{-2}$ is a rational function, despite it was not given by a rational expression in the first place.
However, this is only one view on the topic. Usually equivalent terms like $1/x^{-2}$ and $x^2/1$ are considered as the same term (which is not correct formally, but makes mostly no difference in applications). Hence we say that $1/x^{-2}$ is a rational expression, despite it is only equivalent to one.