This is not an answer, but a request for clarification.  By "multi-digraph" you mean a graph where multiple directed edges can share given starting and ending vertices.  By "chain" you mean a path where all of the arrows on the directed edges are pointing in the same direction.  By "certain subgraphs" you might mean, say, a K4 where the direction of edges is neglected.  By "limits" you might mean, say, that no vertex has more than three directed edges originating from it or more than seven terminating on it; and that no more that four edges can be in parallel between any two vertices…. Is that the gist?

Yes the graphs can have parallel edges that both start and end at the same vertices.

By 'chain' I mean an addition chain. That's just a list of integers with the typical properties of an addition chain. We try to prove an addition chain exists of a specific length and if that fails we try a larger length.

By 'certain subgraphs' I mean that certain portions of the graph can't contain certain things expressible as a subgraph of the original graph. For example you may not have 4 parallel edges but you can have 3 or 2 and of course 1 (which has no parallel edges). There are other excluded subgraphs as well. The subgraphs excluded are directed as well and edge direction must match.

Yes by limits there are limits on the number of edges that may have the same source vertex. There are more complicated limits like a source vertex can't have more than one distinct set of parallel edges emanating from it. So for example source vertex A could have two or three parallel edges to vertex B but that would rule out anything but a single edge from A to vertex C.