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I understand that each geographical dataset has a CRS, and that in order to overlay them, you need to either make sure that all datasets use the same CRS, or re-project some of them (or use on-the-fly projection).

However, I do not really understand one thing; when I set the project CRS in QGIS, I am setting the CRS in which I wish to view the overlay. Now, if that particular CRS is a projected one, then I am comfortable viewing it on my (flat) screen. However, what if the CRS is geographic, such as WGS84? Do the axes then simply correspond to latitude and longitude?

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Yes, the axes of geographic CRS have degrees North/East as units.

That's the reason why such views look distorted the more to the poles you come. One degree of longitude is very small there, but displayed the same as at the equator.

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