Is a perfect matching's weight less than MST of a metric graph?
Given any spanning tree, here is how to construct a perfect matching with weight no larger. We consider two cases.

Case 1. The tree has some leaf $\ell$ which is an only child of its "parent" $x$. In this case we add $(\ell,x)$ to the matching, delete the vertices $\ell,x$, and continue.

Case 2. The tree has some leaf $\ell_1$ which has a sibling $\ell_2$. Let the common "parent" be $x$. The triangle inequality shows that $w(\ell_1,\ell_2) \leq w(\ell_1,x) + w(\ell_2,x)$. Accordingly, we add $(\ell_1,\ell_2)$ to the matching, delete the vertices $\ell_1,\ell_2$, and continue.

At each point in time, the remaining graph is always a tree, and the removed vertices are covered by a matching. Furthermore, the cost of the matching is always at most the cost of the edges removed from the tree. Eventually all the vertices will be exhausted, and we will be left with a perfect matching whose cost is at most the cost of the tree.