Shortest paths from a source in weighted graph with only one negative weighted edge
No. The negative edge will not always be part of the shortest path. For example, in the digraph

A &lt;==[+1]== B ==[+4]==> C ==[-1]==> D ==[+4]==> E

the shortest path departing from B is the path from B to A, and does not include the negative edge (C,D). 

Even when the negative edge is part of the shortest path, it won't necessarily be the first edge of the path.