Why a path in a causal graph can have edges not all with the same direction? In a fork, A <- C -> B, A and B are independent given C. We can say that A and B are d-separated or the path between them is blocked by C, given C. So information does not flow (why?). In a chain, A -> C -> B, A and B are independent given C. Again, we can say that A and B are d-separated or the path between A and B is blocked by C, given C. However, in a collider, the situation is somehow the opposite. In a collider, A -> C <- B, A and B are independent NOT given C, but if C (or any descendant of C) is given they are dependent.
What exactly is a path in a casual model? A path can apparently have edges not all of the same direction, but why? I read that a path somehow tells us if information flows. Which info? Why does a path would allow information to flow (or not)? In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)? In which sense information flows or not flows along these paths? 
 A: The challenge in DAGs is that a single probability distribution can support multiple possible DAGS.
A path in a causal model represents a relation to the "do" operator. If you "do" A, what is the potential outcome of B or C? If the probability distribution for either node shifts as a result of "doing" A, then there is a causal path between them. This is all covered in "Causality" by Pearl.
The notion that a causal path shows a "flow of information" is vague and non-specific, and when you consider the statistical definition of information as a theoretical quantity, it's wrong. It's unfortunate that causality has a lot of overlapping nomenclature with probability, but probability and statistics only serve to quantify causal relationships, not identify them. When you draw the DAG, you impose a structure to the probability distribution that allows you to estimate potential outcomes. If you change that structure, the results and inference are completely different. As an aside, this is a common issue in structural equation modeling.
A: Let's assume that effects have many causes. For example, we can have $A$ and $B$ causing $C$, and $B$ and $D$ causing $E$. The causal diagram is below.

If we somehow keep constant (adjust, control, regress, match, select....) the common causes of $C$ and $E$, which is $B$, the changes they will have will come from their other causes ($A$ for $C$ and $D$ for $E$), which are independent, that is, $A \mathrel{\unicode{x2AEB}} D$. That's why you can measure some dependence (due to the common causes), but it will disappear once you adjust for the common causes. This is an example of confounding ($A \leftarrow B \rightarrow C$), and we would have the same reasoning for a chain ($A \rightarrow B \rightarrow C$). If the relationship between $A$ and $C$ are somehow interrupted, they become independent. $A$ causes $B$ that causes $C$. If $B$ is a system and we turn it off, $A$ won't have an effect on $C$, for the only way to cause $C$ was through $B$.
Things get trickier when we have a collider ($A \rightarrow B \leftarrow C$). If the only relationship between $A$ and $C$ is through their common effect $B$, they're independent. But if you adjust for their common child, you get some information (correlation, statistical dependence). One nice example is throwing dice. If I throw two dice, one after the other, it's reasonable to assume each throw is independent of the other. If I ask you to guess what number I got in the first throw, you won't have anything to help you answer this. If I ask about the second throw, the same. If I tell you the result in my first throw, this doesn't help you guess the second throw. So indeed they're independent. However... If I tell you the first throw and the sum of both results (which is caused by both throws, their sums), you can guess the second throw.

What exactly is a path in a casual model?
A path can apparently have edges not all of the same direction, but
why? I read that a path somehow tells us if information flows. Which info? Why does a path would allow information to flow (or not)? In which
circumstances does info flow or not along a specific path (i.e. a
chain, fork and collider)? In which sense information flows or not
flows along these paths?

In this context we're discussing, a path is a set of edges between two nodes that does not contain colliders ($\rightarrow \leftarrow$). It may look counterintuitive to call $A \leftarrow B \leftarrow C \leftarrow D$ a path between $A$ and $D$ or from $A$ to $D$, with edges oriented backward, but it is a path just like $A \leftarrow B \leftarrow C \rightarrow D$.
What is this info? It's a measure of statistical dependence. In some circumstances, this is spurious statistical dependence (in the sense that it is not direct). If I want to measure the dependence between $A$ and $C$ but they are only connected through a confounder $B$ ($A \leftarrow B \rightarrow C$), not adjusting for $B$ will still give me information, but a biased one. It can help with prediction, but intervening on $A$ won't have an effect on $C$ because they're not directly related.
There are sets of rules that help such analyses, but there are also analytical demonstrations. I tried to explain to you in an intuitive way, with daily examples.
A: A path between variables just means that they are dependent.  We call it a path because it is a path—that obeys the d-separation rules—in a causal graph.

have edges not all of the same direction, but why?

You answered this question in your question.   It's because of the d-separation rules.

In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)?

You answered this in your question.  You can review the d-seperation rules if you want a formal definition.

In a fork, A <- C -> B, A and B are independent given C. We can say that A and B are d-separated or the path between them is blocked by C, given C. So information does not flow (why?).

How can information flow from A to B if C were observed?  What more does A tell you about B?  If C's unobserved, information would flow because A tells you about C, which tells you about B.

However, in a collider, the situation is somehow the opposite.

You can reason about this through common sense.  If A and B cause C, then, not knowing C, A tells you nothing about B.  Given C, then potentially A or B may be responsible for C, and knowing that, for example, A couldn't have caused C, implies that B did.  This is explaining away.
