Rememeber also that there was a hot war in the middle of Europe (former Yugoslavia) during the 90s with the US even carrying airstrikes on an European capital (Belgrade).
Obviously this does not mean that European countries should have weak militaries or not show strength. But the threat of Russia is overblown and used to manufacture consent in public opinion for more spending and more EU integration at a time when people already suffer economically and are already squeezed, and growing disatisfied with the EU.
IMHO, the highest risk of violent instability in Western Europe now and in the coming years is not Russia but mass immigration and islamist terrorism at large. And perhaps that's also partly why governments are trying to deflect attention...
> it seems crazy that Russia might start dropping bombs, say, on Copenhagen or London.
Yes, that is totally crazy.