(1) Let's say $T_1$ has $r$ number of edges and $T_2$ has $s$ number of edges; where $r \ne s$ but two trees has same weight since they both are MST.
I kind feel this kind not be true because to span $n$ vertices with minimum number of edges, we need $n-1$ edges. so MST must have at least $n-1$ edges; if some MST has more than $n-1$ edges, then there must exists a cycle which leads to non-MST.
However, I don't know how to show that if the number of edges is greater or equal to the number of vertices of an an undirected, connected graph $G$, then $G$ has a cycle.
What if the graph is directed? Is this hold as well?