Understanding connection between minimum spanning tree, shortest path, breadth first and depth first traversal
In fact, a MST is a subset of the edges building a tree (thus without cycle) using the minimum edge weights. It has exactly N-1 edges for a N nodes graph. 

It may exist several different MST for the same graph. For an unweighted graph, which is in other way a graph with weight 1 on every edges, the total weight of any spanning tree is N-1. Thus any spanning tree is a MST.

When you use BFS or DFS to explore a graph, you build an exploration tree, which is the shortest path tree for BFS. And if the graph is unweighted, you can say this tree is a MST, yes so 1. and 2. are true.

3. is true, MST and SPT are different. Just make BFS and DFS on a 3-clique (3 nodes connected to each other) to have an example.

4. is true but remember that "minimum" adds no information to any spanning tree of an unweighted graph. And also that a N nodes tree has exactly N-1 egdes.