For A, assume that any subset of edges that connect all vertices and has minimum total weight is not a tree, i.e it consists a cycle, since it contains a cycle, so you can remove the edge which is creating the cycle and still cover all the vertices , and the total weight also decreased, hence our assumption that the sum of the weight of the subset of edges which we initially chose was minimum is false.
There is a contradiction.
$\therefore$ ,we can say that any subset of edges that connect all vertices and has minimum total weight is a tree.
For B, this is what is the invariant for Dijkstra Algorithm, this is also true. You can see the proof of correctness of Dijkstra algorithm from any book.