Let $X_k$ be independent variables with $\mathbb E(X_k)=0$ and $\operatorname {var}(X_k)=1$ Assume there exists $0<C$ for all $k$ we have $\mathbb E(|X_k|^3)<C$ [bounded 3rd moment] Does Central Limit Theorem holds here?
If yes, where can I find a proof? If no, any guidance to building a counter example?
The conclusion of CLT here follows from the Lyapunov CLT. This is essentially the simplest generalization of the classical CLT, and it uses information about higher moments of the $X_k$. You assume that $E[X_k]=\mu_k$ exists and $\operatorname{Var}(X_k)=\sigma^2_k$ exists, and also that $E[|X_k-\mu_k|^{2+\delta}]=t_k$ exists for some $\delta>0$. You then introduce $S_n^2=\sum_{k=1}^n \sigma_k^2$ and $T_n=\sum_{k=1}^n t_k$, and you need to be able to control $T_n$ by $S_n$ in the sense that:
$$\lim_{n \to \infty} \frac{T_n}{S_n^{2+\delta}} = 0.$$
(It is easy enough to remember the exponents: they are set up so that the ratio is dimensionless.) In your case $\delta=1$, $t_k \leq C$, and $S_n^2=n$, so that it is enough to observe
$$\lim_{n \to \infty} \frac{Cn}{n^{3/2}} = 0.$$
The Lyapunov CLT is a special case of the more general Lindeberg CLT, which is essentially "the" CLT for the independent, non-identically distributed case. Lindeberg avoids higher moments by making a more detailed hypothesis about the smallness of the tails.