Do we have $\|f\|_\infty\leq |f(0)|+\|f'\|_\infty$? Do we have that for any $C^1([0,1],\mathbb{R})$ function $f$ this: $\|f\|_\infty\leq |f(0)|+\|f'\|_\infty$?

Yes.
Proof. Let $x\in[0,1]$. By the Fundamental Theorem of Calculus,
$$\int_0^xf'(x)\;dx=f(x)-f(0)$$
and thus
$$|f(x)|\leq |f(0)|+\int_0^x|f'(t)|\;dt\leq |f(0)|+\|f'\|_{\infty}\int_0^11\;ds=|f(0)|+\|f'\|_{\infty}$$
Taking the supremum with respect to $x$, we get the desired inequality.
