I've been studying this topic, but there are a few questions I cannot find the answers to.
When must an edge of a connected simple graph be in every spanning tree for this graph?
For which graphs do depth-first search and breadth-first search produce identical spanning trees no matter which vertex is selected as the root of the tree?
Your comment that an edge which is necessary for the graph to be connected must be in every spanning tree is correct. In fact this is the only reason an edge can be in every spanning tree: if the edge isn't necessary for the graph to be connected we can remove it, and the remaining graph is still connected so it still has a spanning tree.
For the second one, one way it can be true is if there is only one spanning tree. What graphs have a unique spanning tree?
Suppose $xy$ is an edge which is not in every spanning tree. Try to show that (for some ordering of the vertices) BFS uses the edge $xy$ and DFS doesn't.