I have a complete edge-weighted digraph, where the length of every cycle is null, ie. the sum of edges' weights is null. I can see that on each edge the weight in one direction is the opposite of the other one, but it is not particularly useful.
Does this type of graph have a name? Are there interesting properties, beside the ones induced by the completion?
For example, in a somehow similar situation in fluid mechanics, under "classic" hypothésis : $\oint_S \vec{v}.\vec{n}dS=0$ implies $\nabla.v=0$.
NB : the weights are elements from $C^{\infty}(\mathbb{R})$, but any relation with scalar weights is more than welcome since it could probably be transposed.
Sure. You've almost answered your own question: you can pick any vertex $v$ and assign it a "potential" $0$, i.e., define a function $p$ with $$ p(v) = 0 $$ Now for any vertex $w$, you can define $p(w)$ to be the sum of the weights along any directed path from $v$ to $w$. (You can choose any path because of your cycle condition, just as in the fluid dynamics case.)
This defines a "potential" $p$ on the entire graph, with the weight between any two adjacent vertices $P$ and $Q$ being $p(Q) - p(P)$.
I'm not sure what graph theorists call such a graph, but I'd personally be inclined to call it a "gradient" graph, and would start almost every proof with the words "Let $p$ be a potential for the graph $G$; then ..."
Apparently such things (even in the case where $G$ is not a complete graph) are called 'potential difference graphs' is electrical engineering; that's hardly surprising, I guess.