Finding MST after adding a new vertex
As stated in your post, the idea is to use Prim's algorithm with only the edges from $T$ and the new edges, let's call them $E'$. 

For the sake of simplicity, let's assume that $T$ is the unique MST. (This is not required, but it is easier to reason about correctness in this case.)

Now think about what would happen if we ran Prim's on the full graph after the new vertex and edges had been added. We would start with an arbitrary single vertex and successively add edges that are adjacent to the growing tree. 

Here's the key idea: each edge that is added by Prim's on the full graph will either belong to $T$, or to $E'$. If at some step we tried to add an edge in $E$ but not in $T$, then $T$ would not be a (unique) MST.

Since we know that regular Prim's would only add edges from $E'$ and $T$, these are the only edges that need to be added to the priority queue.

As Hendrick Jan said in comments, it is certainly not the case that $|V|-1$ of the edges must be from the old graph, for all of the new edges may be less than the minimum weight edge in the old graph, in which case the MST would include all of the edges in $E'$.