is it possible to express $y$ in this expression:
$y^{x} = e^{\sin x}$
through natural logarithm or, maybe, something else, so we get rid of $y$ at all?
I need to have "something in power of $x$" instead of $e^{\sin x}$ , but I am not sure if it is possible at all.
The value of $y$ can be computed from the value of $x$ if and only if $x\ne0$. So you can safely raise to $1/x$ both sides.
$$ y=\exp\Bigl(\frac{\sin x}{x}\Bigr) $$
Then this function can be extended by continuity at $x=0$.
If you want to compute $$ \lim_{x\to0}\frac{e^{\sin3x}-e^{\sin x}}{x} $$ do first $$ l_k=\lim_{x\to0}\frac{e^{\sin(kx)}-1}{x} $$ by substituting $y=e^{\sin(kx)}-1$, so you get $$ \lim_{y\to0}\frac{ky}{\arcsin\log(1+y)}= \lim_{y\to0}\frac{ky}{\log(1+y)}\frac{\log(1+y)}{\arcsin\log(1+y)}=k $$ Then your limit is $l_3-l_1=3-1=2$.