Statement 1 is true. Proof can be found easily.
The biggest confusion in Statement 2 is what is a cut? Is it cut set, or cut vertex, or cute edge?
Actually, a cut is anything that creates a partition, or "cut" in the set of vertices $V$ of a graph.
A cut $C=(S,T)$ is a partition of $V$ of a graph $G=(V,E)$ into two subsets S and T.
Notice "cut" is actually a set of two subsets.
One might say that cut is a cut set.
So, if for every cut set of a graph, there exists a unique minimum weight edge; then graph has a unique MST. True.
Why? Because the whole graph can be seen as the collection of these cut sets and from each cut set we can only pick the unique minimum weight edge.
Option C; both the statements are true.
Let G be a connected, undirected graph. A cut in G is a set of edges whose removal...
These wordings are taken from GATE 1999, Question Number 5.