Is a tournament graph the same thing as a directed complete graph? And, do all vertices in a tournament graph have the same number of edges?
Unless I'm missing something obvious then the answer to both your questions is "yes"
A tournament graph is defined as a complete graph with a direction chosen for the edges. Hence it is a directed complete graph.
Wikipedia definition of a Tournament Graph
and since a complete graph has an edge between each vertex, then every vertex has the same number of edges, and this doesn't change when a direction is chosen for the edge.