Is Dijkstra's algorithm used in modern route-finding systems such as Google maps or the satnav in your car? If not, then what is?
Yes, Dijkstra's algorithm is used in modern maps systems. A lengthy and informative discussion can be found in the following question from StackOverflow: What algorithms compute directions from point A to point B on a map?
Since then, some mind-blowing methods have been discovered, capable of doing cross-country routing in fractional milliseconds - the so-called "two-hop labeling distance oracles". See here, or search for "Hub labeling" or "Shortest paths for the masses". I think I heard Bing uses this one. It also has applications like the ability to find the nearest point of interest (e.g. nearest gas station) with a complexity independent on the number of gas stations.