A Complete Digraph is an Undirected Graph? Can I consider the Undirected Graph as a special case of Digraphs where all edges points for both directions?
A complete (completely connected) digraph turns to an undirected graph?
 A: This is often, but not always a good way to apply a statement about directed graphs to an undirected graph.
For an example where it does not work: plenty of connected but undirected graphs do not have an Eulerian tour. But if you turn a connected graph into a directed graph by replacing each edge with two directed edges, then the resulting directed graph will always have an Eulerian tour. 
The problem in this example is that an Eulerian tour in the undirected graph can traverse each edge $vw$ only once, whereas an Eulerian tour in the directed graph we obtain would have to take the directed edge $(v,w)$ once and the directed edge $(w,v)$ once.
On the other hand, connectivity is preserved: having a path from one vertex to another in the undirected graph is equivalent to having such a path in the directed graph constructed from it.
Whenever you have a tool for dealing with directed graphs, you can always try to apply it to an undirected graph by using this double-edge construction, but you should check to see what the effect is before proceeding.
