Why is the notation for differentiation like this? Consider the notation for denoting the differentiation of a function $f(x)$. 
$$\frac{d[f(x)]}{dx}$$
I mean, this notation doesn't make any sense. $dx$ means a vanishingly small $x$, which can be understood, if we take $x$ to mean $\Delta x$ in a loose sense. But what does $df(x)$ denote? 
I mean, ideally, shouldn't differentiation be denoted as
$$\frac{f(x_0 + d\Delta x) - f(x_0)}{d\Delta x}$$
Is this done just to simplify things, or is there another reason? 

It's a shorthand: "the derivative of a function, $f$, which is a function of $x$, with respect to $x$."
Your second definition would be
$$\left.\frac{d\left[f(x)\right]}{dx}\right|_{x=x_0}$$
if you take the limit of that expression as $\Delta x \to 0$.  Otherwise, it's more of a finite difference than a differential.
