Does the zeroth root exist? Definition of $Nth$ root:
$3rd$ order inverse group $1$ hyperoperation.
Division is how many times you can subtract a certain divisor from the dividend before it becomes negative.
Likewise Nth root is the result of repeated division by a certain divisor before it becomes $1$ or a decimal. The number of times you divide it before it becomes a decimal is the index.
Ex: $\sqrt [3]{8} = (8/2)/2$
Is the zeroth root even defined and if so what is $\sqrt [0]{x}$

No this is usually not defined. One definition of the $n$th root is
$$
\sqrt[n]{x} = x^{\frac{1}{n}}.
$$
So for a fixed $x>1$ you see that 
$$
\lim_{n\to 0^+} x^{\frac{1}{n}} = \infty.
$$
