In an undirected graph, can two nodes at an identical distance n from the root of a DFS tree be neighbors in the original graph? I'm thinking no, but I'm not sure (because of back edges)
Your question does not stipulate how the DFS tree is generated. Naive implementations simply list nodes in the order they are visited, but most practical implementations rebuild the output tree as the search progresses and the ranks of the nodes are adjusted.
Consider three nodes, A B C, connected to each other. A naive DFS will visit A B C, with ranks of 0 1 2, and the tree will have a back edge from C to A. A more useful implementation will backtrack from C to B to A, and then from A will travel to C again and assign it a rank of 1, then decline to travel to B at rank 2 because B is already rank 1, this tree will have both B and C at rank 1 below A, with a cross edge between B and C.