Statements in the Question must be written like this :
A. if $e$ is a specific edge with minimum edge weight in a connected weighted graph,it must be among the edges of at least one minimum spanning tree of the graph.
B. if $e$ is a Specific edge with minimum edge weight in a connected weighted graph,it must be among the edges of each one minimum spanning tree of the graph.
Now, Coming to answer :
$A$ is Correct. Think about Kruskal's algorithm. It randomly picks the Edge with minimum weight. So, If there are multiple edges with minimum weights, then it can pick any of them, Hence there will always be some MST for each Minimum weighted edge(i.e. Edge with minimum weight).
$B$ is False. Consider a Triangle(Cycle graph with Three Vertices) with each edge having weight $1$. Of, Course, in Each MST, One edge will definitely be missing. So, It is NOT necessary that a Edge with the minimum weight will be present in Each MST.