Want to put a disclaimer, I am not a military buff, and openly will admit a lack of consuming foreign policy "expertise" or analysis or anything else. So is far more and worse than average armchair hypotheticals, wedded more in my historical knowledge I acquired while polishing bowling balls. And also I feel this is an instance where my anarcho-pacifism may be blinding me to grim realities.
Nintex, your last part (and earlier handicapping of the nations' potential dangers) there is kinda key and something that's been talked about in various IR circles off and on. The resources and capacities for wars as we once knew them don't seem to exist.
The "great powers" don't engage in anything resembling wars of that scope. Russia's adventures in places like Crimea and South Ossetia were essentially invitations by one side of a dispute or low level civil war, the same was true of Chechnya and their other interventions in the Caucuses. The Iraq War, Afghanistan War and most of the rest of the War on Terror operations have had almost battles for show with standard military forces. UN interventions of the past two decades have been disasters launched from the air nearly exclusively.
If you discount Africa, for multiple reasons, you could make an argument that the only nation to show the ability to fight a "true war" against anything resembling formal opposition this century is Israel. And that's a seriously lower weight class. With Pakistan as maybe a runner up due to combined experience against so many foes and their best buddies India running third.
The United States is the only confirmed power that can engage in a large scale war for an extended period. Russia and China's power is still on paper (with many of China's conflicts having similar situations to what Russia has been engaged in) and there's good reason to doubt both, especially once extended. India, Pakistan and Israel are all constrained by geography, and Israel rather infamously generally deplores extended commitments which is why the 2006 Lebanon disaster led to some strategic soul searching. Turkey is constrained in a lot of ways.
A Russia expansion can only look West, which is a war they lose on paper. The Russians historically have only really ever gone right up to the edge of the Middle East and said "nah, we're good here." Preferring proxies, except there's not exactly any in the entire region right now. They're all seemingly dependent on the U.S. too much or in utter chaos. Or both.
An economic collapse that takes a sledge hammer to the Turkish, Russian and Chinese economies would kneecap extended operations far more than it would the EU-US alliance. Especially regarding logistics. The US-EU has too much legacy hardware that's also close enough to modern, it also semi-surprisingly has the far larger in service and active manpower base, with both active and reserve units having recent modern warfare experience. (Technically, this is not true, but I have serious doubts regarding the capabilities of the North Korean Army, especially to deploy outside the peninsula, no matter what THQ tried to make us think.)
Not saying a First/Second World scale war couldn't be attempted and wouldn't suck balls, just that even in a serious economic downturn, I don't quite see it as a serious consideration by true powers.
Maybe something like Turkey stupidly trying to march into Syrian gap and also deal with the Kurds seems semi-plausible. Though interestingly, if these statistics I found are accurate and they haven't lost them recently to ISIL, Syria has an absurd number of tanks, random armored vehicles and older artillery that must be serious hand me downs for a country of its size, population, etc. Along with decent enough basic air support (vs. Turkey) that my strategy would be to pile these up along the border (plus any old cars and other worthless crap) and then bomb Turkish attempts to get through the wall. Yes, I have sometimes used similar tactics in Civ with outdated units, especially in V and VI's one unit per tile system, why do you ask?
