Can BFS be used to check if a graph is Bipartite (2-colorable)?
Yes.
Run BFS. Color layers alternately (Layer 0=Red, Layer 1=Blue...). If you find an edge between two nodes of the same color, it is not bipartite.
Does BFS find the shortest path in a weighted graph (even positive weights)?
No.
BFS only counts # of edges. It treats edge weight 100 same as edge weight 1.