Existence of a route following one-way streets
First, it definitely won't result in a DAG, just a digraph. You need to get from A to B and B to A in order to satisfy your condition for nodes A and B, but that itself is a cycle.

Lots of time BFS and DFS are equivalent in relation to a problem, so it doesn't matter which one you choose. Here, the real trick is not choosing between DFS and BFS, but dealing with the fact that the graph is directed.

If you don't want a direct answer, here's the strategy I would use: first find a representation for the streets in terms of a digraph. What corresponds to edges? What corresponds to nodes?

Then, you need to find a graph algorithm that will work for this problem. What information do BFS and DFS give you about the graph? How can they be adapted to a directed graph?

To give you a little hint, I'd recommend you Google Eulerian chains.