My point concerning nuclear deterrence is that it negates full scale war in Europe. This is undeniable insofar as it negates the necessity of having a military capable of fighting a full scale war in Europe. Your point regarding skirmishes, interventions, and other small scale subversions of the international order, however, remains valid.
That said, Europe is supremely unconcerned about North Korea, the Taiwan Strait, and Sino-Japanese grandstanding. Again, for the foreseeable future the role of world police is almost exclusively American. I don't see why Europe would have any issue with that, since American cultural identities and values are quite European-aligned and thus non-threatening. This restricts Europe's area of interest significantly, and further reduces the need for intervention capacity.
Still, you are severely underestimating European power projection. While Putin is 'making fools of Europe' making a show of submarines and fighter jets, Russia's economy has just imploded. Its like bread and circuses, but they're running out of bread. Except for Ukraine, these shows are largely irrelevant to European interests, hence the 'impunity'. If Argentina were to make good of their saber-rattling in the Falklands they'd get a whooping just like last time. The UK has the most active military in Europe because it cooperates with the US so often, in Afghanistan, Iraq, etc. Apart from the US and Russia, every operational carrier group is European. The UK has two carrier groups. France, Italy, and Spain also have one each. Every other nation has old European handouts for training purposes. The recent intervention in Libya was done mainly with EU assets. Europe is pretty capable of defending its peripheral interests militarily if needed be.
Finally, about beliefs. A nation state is about some sort of national unity, and about some sort of state structure. European countries are particularly resilient because most are pretty strong at both. See Germany, who not only exists after multiple total fuckups, but is doing pretty well. French national unity has survived multiple failed states as well. Likewise, most states that have their ideologies threatened do not 'die', they simply adapt to new belief systems. The regular people, of course, mostly just go about their daily lives regardless.
Rome did indeed fail at transitioning badly enough to 'die', for multiple reasons. The Soviet Union? I'm not so sure it died, considering its still controlled by the same oligarchic elites as before, but with another name and a token democratic process. Russia is not an empire with that many satellites anymore, but neither is France and the UK and they also seem to be doing okay. Adapting to Islam should not be that big of a deal, since most liberal states nowadays are quite secular and multicultural. Sure, Islamic extremism is quite problematic (a civil problem, not a war problem), but Europe has a long history of religious conflict. Usually the state 'wins' and everyone ends more civilised. Hopefully history will repeat itself in this way as well.