My question is : do mathematicians have a concrete/visual mental picture of the (sometimes very abstract) concepts they manipulate, which can help them finding new leads/paths/theorems ?
Speaking from my experience, I think most have.
Whenever I doing differentiation, I always imagine "zooming" into the graph, to see it's locally a plane... With this intuition I discovered many laws about differentiation (and other geometric stuff) before I learned about them. The latest example was the formula for the total derivative, so when you have a function $f(t) = g(x(t), y(t))$ and you want to get the $df/dt$.
When working in 4 higher dimensions or higher in the simplest cases making Schlegel diagrams (so in 4 dimensions simply dividing dividing $x$, $y$ and $z$ with $w$ ans imagine the resulting picture) or simply projecting into 3D, by omitting coordinates often work. But more often it's not working. In this case I try to find dimensional analogies. For example no one can directly imagine 4 dimensions. But still we can measure distances, because the formula in 1D is $\sqrt{\Delta x}$, in 2D it's $\sqrt{\Delta x + \Delta y}$, in 3D it's $\sqrt{\Delta x + \Delta y + \Delta z}$. One can easily continue it in 4D and have the formula $\sqrt{\Delta x + \Delta y + \Delta z + \Delta w}$.
When you become accustomed with the concepts, you will need to imagine less, and simply use the laws you learned. Then you will develop an intuition when you can simply "feel" where is the right path, or whether the results you got makes sense or not.
In some of the answers I received on previous posts, I've been confronted to very abstract concepts which I find difficult to grab, because I don't see any physical meaning to them.
It depends on the concept. Sometimes you simply cannot connect it to a physical meaning for example: prime numbers.
Also the problem I see is that writing about math in understandable way requires good writing skills, that only a few have.
So when learning something new I also struggle a lot with papers/textbooks where definitions and theorems work with whole fields, whole sets, $n$s and $m$s. The definition and theorem is impenetrable, no one can nitpick. But I think accompanying it into a plain English sentence would greatly increase the chances to understand and see the concept, even if it's not precise enough.
For example: matrix determinant has a difficult and artifical definition, but telling something about parallelogram areas and parallelepiped volumes would help a lot.
Or another example: a the graph of a differentiable function generally don't have cusps or edges.