Select the n closest nodes from a starting node in a weighted directed graph
Question 1. For very large graphs, the canonical Dijkstra algorithm will not work because you must enqueue every node first. 


Yes, you are right about the disadvantage of canonical Dijkstra's algorithm. However, it can be easily adapted (wiki):


  The algorithm can start with a priority queue that contains only one item, and insert new items as they are discovered (instead of doing a decrease-key, check whether the key is in the queue; if it is, decrease its key, otherwise insert it).


See the paper for details (including experiments) on implementing Dijkstra's algorithm with such a modified priority queue.


  Question 2. I'm interested in things like how do you know when you've found all the shortest paths for the closest nodes, without traversing the entire graph in the process.


The Dijkstra's algorithm finds the shorted paths from a starting node in the order of increasing costs. So, the first $n$ vertices removed (delete-min) from the priority queue are the $n$ closest nodes from a starting node you are looking for.