Minimum spanning tree with two minimum edge weights
Yes $G$ has a unique MST (assuming it is not a multigraph). Take a look at Kruskal's algorithm: the edges are ordered ascending by weight and then added to the MST unless there is a cycle. After adding the two minimum edges, you can't have a cycle, therefore both are in the MST.

Edit: As Paresh pointed out, this is not a complete proof, since there may be MSTs which cannot be produced by Kruskal's after permutating equally weighted edges in the sorted list. But Kruskal's is in fact able to produce any MST. Proof.