• Yes, "shortest path graph" defined for a particular source is a DAG.
• Let's call the required edges "special edges". Let's also consider only the subgraph $$$G$$$ of shortest path graph which includes only the nodes which can reach $$$t$$$ in the shortest path graph.
  
  Now special edges are essentially edges through which all paths from $$$s$$$ to $$$t$$$ in $$$G$$$ pass.
  
  How do we check if an edge $$$E : (u\rightarrow v)$$$ in $$$G$$$ is special? There is just one condition:
  
  
  Spoiler

  Consider all edges (except for $$$E$$$) $$$(x \rightarrow y)$$$ in $$$G$$$ such that $$$dis(s, x) <= dis(s, u)$$$. Then $$$max(dis(s, y))$$$ over all such $$$(x \rightarrow y)$$$ must be $$$\leq dis(s, u)$$$.

  This can be easily be checked by maintaining the two largest $$$dis(s, y)$$$ encountered.

• Analogously defined vertices too, can be found in a similar manner. When checking for some vertex $$$x$$$, just check following:
  Spoiler

  1. There should be no other $$$y (\neq x)$$$ such that $$$dis(s, x) = dis(s, y)$$$.
  2. For all vertices $$$z$$$ such that $$$dis(s, z) < dis(s, x)$$$, there should be no outgoing edge $$$(z \rightarrow w)$$$ such that $$$dis(s, w) > dis(s, x)$$$.

So both the problems can be easily solved in $$$O(n\log{n})$$$.

In the shortest-path graph all edges are directed away from S and towards T, so if an edge is a bridge when you undirect the edges it will also be a bridge in the DAG.