Is it logical to attempt differentiation of y=1? Is y=1 different from y=x^0? I am wary of getting in over my head, after initial searching it is apparent to me that I don't know how to properly phrase this question for the answer I want. I am only working at high school level but in class we learnt differentiation and how when 
$y = ax^n$, 
$ \frac{dy}{dx} = anx^{(n-1)}$.
I was wondering if this applies to when
$y = 1$,
$\frac{dy}{dx} = 1\times1^0 = 1$.
and also when
$y = 1^2$,
$\frac{dy}{dx} = 2\times1^1 = 2$.
Obviously the gradient should be 0, and when calculated in terms of x it makes sense
$y = x^0$,
$\frac{dy}{dx} = 0\times \frac{1}{x} = 0$.
I was wondering how this should be approached, since to me it implies that you CANNOT have a line without a $y$ AND $x$ term, yet $y=1$ CAN be drawn.
Is y=1 a different thing to $y=x^0$? Does $y=1$ even exist in 2D space? Or do you just have to simplify your equation before differentiating it?

In calculus you differentiate functions, not formulas. Much of your confusion comes from thinking about variables $x$ and $y$ instead of functions. But since that's how you're learning I will try to use that vocabulary. 
The function you have in mind when you write $y=1$ is the function whose value is always $1$, independent of the value of the independent variable which you are implicitly thinking of as "$x$". The graph of that function is a horizontal line of height $1$. Its slope at every point is $0$, so the derivative of the function is $dy/dx=0$.
Some functions that you encounter have formulas. For example, the formula 
$$
y= x 
$$
has graph a straight line sloping  up from the origin while 
$$
y= x^2
$$
describes a parabola.
Early in the study of calculus you show that when a function is given by a formula
$$
y = x^n
$$
for a positive integer $n$ then its  derivative  is given by the formula
$$
\frac{dy}{dx} = n x^{n-1}.
$$
Later in your study you will show that formula works for all values of $n$ (not just integers) except for $n=0$. In that case you know the answer right away because the intent it to describe a constant function, whose graph is horizontal. The formula just happens to give the the right answer except when $x=0$, but you shouldn't use the formula. That's the long answer to your good question.
