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)?
 A: 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.
A: 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.
A: 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 $\mathbb R$.)
A: The mean is the point where the area left and right between the CDF and the vertical line through the mean are equal.
For symmetric distributions, the left and the right side/area have the same shape and you get that the mean is at the point where the CDF is equal to 0.5 (also the median).
For non-symmetric distributions the shapes are different, so this equality of area does not need to be true (obviously it can be equal, but it does not need to be equal).
For any distribution where the mean and median is different, you have that the CDF of the mean is not 0.5.
An example is a non-central t-distribution which is plotted below.

