Your inverse monoid without the partial sum relations seems like a variation of McAlister's monoid [here][1] but for paths in a graph instead of words over a set.

The main difference is that the McAlister monoid is the case the graph is a bouquet of loops at a vertex and satisfies just 1,2.  I suspect if you take the tight algebra in the sense of Exel of the McAlister monoid it would impose your relations (3).  One should check that though.


  [1]: https://ac.els-cdn.com/S0021869397973014/1-s2.0-S0021869397973014-main.pdf?_tid=2c958ec1-ed8e-4e03-b119-ed7302662dbd&acdnat=1522084399_8160c3d8c4cc28c11587b91f24bcf4c8

If you forget about property (3), which I think will come from going to the tight algebra, non-zero elements of you inverse semigroup can be represented by directed paths in your graph together with a distinguished in vertex of the path and out vertex of the path (they don't have to be the first or last vertex and they can be the same).  You can multiply if you can line up the out vertex of the first path with the in vertex of the second and take the union and get a valid birooted path.  

I believe that the empty path at a vertex gets identified with your two sums in the with tight algebra although maybe if there are sources or sinks one has to be a little careful.