If $\lim\limits_{x \to \pm\infty}f(x)=0$, does it imply that $\lim\limits_{x \to \pm\infty}f'(x)=0$?

No. Try $f(x)=\sin(x^a)/x$ for various values of $a$.

Key fact: on small scales, you can change the derivative of a function hugely without changing its values very much. I can introduce a tiny wiggle in a curve that is barely noticeable in size, but very sharp in derivative.

Throw in a few tiny wiggles arbitrarily far along a decaying function of your choice, and you'll get a fucntion which decays but whose derivative keeps spiking.

As a more concrete hint, consider a differentiable function $f$ such that $|f(x)| < 1$ for all $x$, and take $g(x) = \frac{1}{2}f(2x)$. Then $g(x)$ has $|g(y)| < \frac{1}{2}$ for all $y$, so $g$ takes smaller values than $f$, but $g'(z) = f'(2z)$, so the derivative of $g$ gets just as big as the derivative of $f$.

This shows how very small functions can nonetheless have very large derivatives.