Is the CDF of the Mean always 0.5 for all kind of distributions? Can we say that the value of the cumulative distribution function at the mean F(X< Mean) is always 0.5 for all kind of distributions (even ones that are not symmetric)?

No.
A counterexample will be $X \sim Exp(1)$. The mean of $X$ is 1, and $P(X < 1)$ is $1 - 1/e$, around 0.63.
