Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?
The answer to the question in the title, "Is a set of acyclic $|V|-1$ light edges always a Minimum Spanning Tree?" is no. That is, the second claim in the question is not true. Hence, the third claim is not true, either.

Here is a simple counterexample. Let $H$ be a graph the complete graph with 4 vertices, $\{a,b, c,d\}$ and weights, $ab\rightarrow 1$, $cd\rightarrow 1$, $ac\rightarrow 2$, $ad\rightarrow 2$, $bc\rightarrow 2$, $bd\rightarrow 2$. The weight of a minimum spanning tree $\{ab, ac, cd\}$ is $1+1+2=4$. Consider the spanning tree made by the 3 edges, $\{ac, bc, bd\}$, each of which is light for the cut $\{a, b\}, \{c,d\}$. However, the weight of this spanning tree is $2 + 2 + 2 = 6$, which is greater than $4$.

One might be wondering why it might be so hard to come up with a counterexample. Part of the reason might be that if the weights of edges of $G$ are different, then $G$ has only one minimum spanning tree, which includes all light edges for every cut.