Graph Algorithms

Let $G = (V, E)$ be *any* connected undirected edge-weighted graph. The weights of the edges in $E$ are positive and distinct. Consider the following statements: (I) Minimum Spanning Tree of $G$ is always unique. (II) Shortest path between any two vertices of $G$ is always unique. Which of the above statements is/are necessarily true?

The Breadth First Search (BFS) algorithm has been implemented using the queue data structure. Which one of the following is a possible order of visiting the nodes in the graph below?