What gives coffee its acidic (sour) taste? I read somewhere that too cold water is what gives coffee its acidity, how does that take place?

Furthermore, does that apply only to the water during the brewing process or as my cup of coffee cools down, the acidity increase?
That's correct: as your cup cools, perceived acidity increases. This is due to variation in the chemical species affected by temperature. 

Coffee is like ice cream. Ice cream is least sweet when it's stone cold, and tart and heavy when it has melted. Similarly, coffee is sweetest (and best) while it's hot, and is bitter or acrid when it has cooled. Furthermore, the cooler temp has led to chemical degradation and thus the awful taste. However, you won't perceive its flavors when the coffee is too hot -- so let it cool a bit.

As for your question whether the on-contact brew temperature affects end-result sweetness-sourness: there are a lot of baristas and coffee R&amp;D pros who do temperature-control brewing, especially for espresso, in order to modify the flavor of the coffee. 

I believe that varying temperature affects coffee flavor in a gradient.

For cold brew (in case you're also curious about that), the cold temperature is a different environment entirely. Coffee is just as soluble, however less energetically kinetic, so more time is substituted for the lack of temperature. The end result is a brew product that functions naturally as a cold product -- nice, sweet, characteristic.