Darwin Proven Wrong?

[Tonus]

When biologists talk about an "evolutionary arms-race," do they in any sense mean that otherwise generally random gene mutations somehow directly respond to the competition?

I think it's more the other way around. A predator may develop a particular feature that gives it an advantage over a particular prey. That feature will influence the evolution of that prey animal by selecting out the ones that don't evolve an effective counter-measure, so to speak. The prey species did not develop a counter-measure all at once; some did and some didn't, and those who did were more likely to survive.

Simple example, if a species of lion develops the ability to run a bit faster and farther than other lions, natural selection will favor them because they're more likely to capture prey for food and thus more likely to breed and pass along the genes for faster, longer chases. By the same token, those among its prey who develop greater speed or endurance will also be more likely to survive and breed and pass along their genes.

But also remember that there are a lot of extinct species in Earth's past. I think that most of the time, the "arms race" is not much of a race at all. Maybe a particular species of gazelle died out before it was able to evolve the features that would have allowed it to survive. Maybe there are even species of predator who managed to wipe out their primary food source and died out before they could adapt to new food sources or evolve to hunt down others.