I've recently gotten a Cafe Press for making my coffee, and I'm pretty bad with proportions. If I make too much and leave it sitting in the press with the grounds pressed to the bottom, does the coffee on top keep 'brewing'? Does it otherwise adversely affect the flavour? Does it affect the caffeine content?
The brewed coffee stays in contact with the ground, even though you pressed the sieve to the bottom. It will eventually release the unwanted flavors it contains, albeit slowly. The ground coffee should be in contact with the hot water for about 30 seconds.
See this answer for the caffeine content.