Hi Joseph,
It seems like there's 2 main questions here, so I'll write them out below with answers.
1) How does network analysis deal with inputs which are not directly located on a network edge/junction?
When loading inputs into a network analysis layer, network analysis objects are created for each input. The network analysis object has it's own network location, which is it's position on the network. If the object isn't directly on a network feature, then the analysis layer will do a search to find the closest network feature and kind of snap to the feature. The proportional location of the point on the line becomes the network location. Check out What are network analysis objects?—Help | ArcGIS Desktop for more information on network analysis objects and network locations.
2) Is the Location-Allocation analysis using Dijkstra's algorithm?
The LA solver uses an OD Cost Matrix as part of it's calculation, and the OD Cost Matrix uses an algorithm which is based on Dijkstra's. You can read more about the algorithms used in network analysis here Algorithms used by the ArcGIS Network Analyst extension—ArcGIS Pro | ArcGIS Desktop
I hope this answers your questions!
-Rachel