Vertex descendants subgraph

You are looking for VertexOutComponent.

VertexOutComponent[g, 4]

gives you the successors of 4. Use Subgraph to get an actual graph out of those. With HighlightGraph, you can also use a subgraph, it will highlight both vertices and edges: HighlightGraph[g, Subgraph[g, VertexOutComponent[g, 4]]].

For visualizing the graph, use GraphLayout -> "LayeredDigraphEmbedding", which will place the root at the topmost position. Some other tree layouts have a "RootVertex" suboptions to achieve this (e.g. for undirected where anything can be the "root").

You could also do for example the following for recovering the result as a graph:

HighlightGraph[g, s = Last@Reap@BreadthFirstScan[g, 3, {"FrontierEdge" -> Sow}]]

Graph@@s