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, this is false. That point is the median, and it is not equal to the mean in all cases, for example, an exponential distribution.
$$
X\sim \exp(1)\\
\mathbb E[X]=1\\
\operatorname{median}(X)=\log 2\approx 0.69
$$
We can simulate this in software, such as R.
set.seed(2021)
X <- rexp(10000, 1) 
mean(X) # approximately 1
median(X) # approximately 0.69

EDIT
The asymmetric Laplace distribution is an example where the distribution extends over the entire real line, not just the positive numbers. (In more technical language, the PDF has support on all of $\mathbb R$.)
