Is 0 divided by a non-zero indeterminate equal to 0.

What you have shown is that assuming $y\neq 0$ the limit is zero but it doesn't suffice.

What matter is that for $y\neq 0$

$$ \dfrac{|x|}{\left(\dfrac{x}{y}\right)^4+1} \le |x| \to 0$$

and since for $y=0 \implies \dfrac{xy^4}{x^4+y^4}=0$ the proof is complete.

There is an easy way: \begin{align*} \left|\dfrac{xy^{4}}{x^{4}+y^{4}}\right|&=|x|\dfrac{y^{4}}{x^{4}+y^{4}}\\ &\leq|x|\\ &\rightarrow 0. \end{align*}