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)?

As others have said, no. In fact, it can be any number in $(0, 1$).
Let $\epsilon \, \in \, (0,1)$, consider the random variable $X$ such that $\mathbb{P}(X = -1) = \epsilon$ and $\mathbb{P}(X = \frac{\epsilon}{1-\epsilon}) = 1 - \epsilon$. The mean is
$$E[X] = -\epsilon + (1-\epsilon)\frac{\epsilon}{1-\epsilon} = 0 \quad, $$
and $F(0) = \epsilon$.
As observed in the comments, $1$ can also be attained by the CDF if we consider a constant random variable.
