I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in C([0,1],\mathbb{R})$. I shall give necessary and sufficient conditions. My answer is: $w$ should only be $zero$ on a null set and otherwise positive. But I guess this is a necessary and also sufficient condition. Am I right about this? My idea is that otherwise, there will be some interval $[a,b] \subset [0,1]$, where a function like $f(x):=e^{\frac{-1}{(x-a)(x-b)}}$ on this interval and otherwise 0 will not fulfill $\langle f,f \rangle \ge 0$. I am somewhat worried about my answer because my sufficient and necessary conditions coincide.
Then I am supposed to tell, when the induced norm by this integral is equivalent to the one induced by $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x) dx $$. My answer is: It is true iff $\inf(w) >0$.
You are close with the condition, but yours is stricter than necessary.
Of course the weight must be non-negative, and it must not vanish on any interval that contains more than one point. But that does not imply that its zero set must have measure $0$, if $F$ is for example a fat Cantor set, and $w(x) = \operatorname{dist}(x,F)$, then $w$ is continuous, vanishes on a set of positive measure, but not on any nontrivial interval. Therefore
$$\int_0^1 \lvert f(x)\rvert^2 w(x)\,dx > 0$$
for every continuous $f$ that does not identically vanish.
If you considered a space of measurable functions, and not of continuous functions, then the zero set of $w$ would have to be a null set.
Your answer for the equivalence is correct.