What's the complexity of calculating the shortest path from $u$ to $v$ with Dijkstra's algorithm using binary heap?
Your answer actually depends on what implementation of Dijkstra's algorithm is used. The textbook implementation inserts all nodes in the queue, and then proceeds with the extraction. The other starts with only the source in the queue. This is generally more efficient in practice, although it needs a bit of book-keeping.

For the standard implementation, you can do better than $O(n \log n)$. Instead of successively inserting each node in the heap based queue, you create an array, and "heapify" it. "Heapify" is a bulk operation for converting a list of numbers into a heap, in $O(n)$, or rather $\Theta(n)$ (see here or here). After that, there will be only one/two extract-min operations (one or two depending on whether the source is originally in the queue or not) in $O(\log n)$. So overall, $\Theta(n)$.

For the other implementation, you can do it in $\Theta(\Delta \log \Delta)$, where $\Delta$ is the degree of the source $u$.