If I glance at my start screen, do I need to wait a few seconds for the tiles to show up-to-date information? Or do they continue to update when I'm not looking at the start screen?
Yes, Live tiles are "live." You can actually check for or disable live updating by right-clicking a tile, waiting for an options bar to slide in, and clicking "Turn live tile off" on an app-by-app basis. The weather app, mail app, and calendar app are quick examples of the automatic updating of Live Tiles.
They will update without you looking. A good example of this is the Finance app where it will update the stock prices on its own.