A person versed in game theory would probably be better suited to answer your first question, rather than a historian. However, in most instances, war comes down to "Us vs Them" because any 3rd party trying to "split it down the middle" and fight 2 other factions would necessarily be at a disadvantage compared to a faction merely fighting 1 other enemy instead of 2, especially since the other 2 factions might put aside their differences temporarily to screw your faction over (like how the Nazis and Soviets divvied up Poland in 1939).

Even when you aren't actively fighting in a war, simply being an independent 3rd faction is liable to piss off one of the two factions fighting in a war. Take for example Denmark during the Napoleonic Wars. They were not an ally of either Napoleon or the British, yet the British burned down their capital and sank their fleet, simply to prevent Napoleon from getting his hands on it. Likewise, the French fought an undeclared naval war against the Americans from 1798-1800 (the Quasi War) simply because the Americans refused to openly support France during the Revolutionary Wars (and also the US reneged on it considerable debt to the French, arguing the debt was owed to the French Monarchy, not the French nation--the cheeky scrubs!).

To sum up, the circumstances where it is sensible and feasible to involve one's faction in a war by allying with neither side and fighting both at the same time are rare. Moreover, throughout most of history, there has been a clear "bigger threat" which induces other, smaller, weaker nations to join forces even when those allies may not have much in common--in fact, they may hate each other.

The classic example is WWII, when the US and UK partnered with the USSR (despite their many outstanding differences) to defeat Nazi Germany, or the previous war when the UK (a liberal, constitutional monarchy), allied itself with Czarist Russia (a near absolutist monarchy and traditional threat to British India) to defeat Imperial Germany.

An example of this "Us vs Them" mechanic from an earlier time would be The War of the League of Augsburg, when England and Scotland (this is shortly before the Act of Union in 1707), the Dutch Republic, the Kingdom of Sweden, the Spanish Empire, the Duchy of Savoy, and the Holy Roman Empire all united to stop France from dominating the European continent, despite most of those allies (most notably the Dutch and English) having fought wars against one another in living memory. But at that moment, stopping the bigger threat was more important than trying to screw over your smaller, more traditional enemy.

For an example of a 3+ faction war though, there is the Russian Civil War, where (for a time) the Reds (the Bolsheviks), the Whites (who were mostly anti-Bolshevik and somewhat pro-Czar), the Peasants or "Green Armies" (who were pretty much a "fuck everyone" faction and fought fiercely against both the Whites and the Reds) were all fighting each other. On top of all of that you have factions like the Cossacks whose loyalties were "flexible" (to put it mildly) and interlopers like the Czech Foreign Legion, people who were allegedly on the side of the Whites but mainly were "We want out of this place and will fight anyone who gets in our way".

There some examples of this in WW2:

In Poland, the Polish Home army fought against both the Axis and the USSR. Polish partisans were certainly no friends of the Germans, and they knew that the Soviets were not exactly liberating their nation either. The Polish partisans also fought against Lithuanian and Ukrainian resistance groups.

You may have seen the Daniel Craig movie "Defiance", based on the partisan group headed by the Bielski brothers. The movie made things very straightforward as they got to gloriously battle against Nazis but in reality they were quite famous in the area for atrocities against local civilians (who were ALSO oppressed by the Nazis).

Yugoslavia was a violent mess during WW2 as well, with various groups fighting each other. The Chetniks, for example, were a major royalist resistance movement while Tito's partisans were communists. They battled each other quite brutally and there were lots of massacres, all the while both sides wishing their country to be free from the Axis.

It helps to remember that a lot of Eastern Europe suffered under both Communism and Nazism, so many of them were not exactly eager to be under the boot of Hitler or Stalin. Some of these people sided with the Nazis to battle the Soviets, or joined up with the Soviets to battle the Nazis, while others of a more hardcore nationalist bent would fight them both!

If you look into the Pacific theater, the war in China had the Nationalist and Communist armies in opposition to each other as well as the Japanese. Besides this, there were other warlords that were not aligned to either the Nationalist nor Communists.

Now it was pretty much unheard of to have an actual three-way shootout going on during an actual battle, but there were certainly 3-way (or more!) wars going on.

This can happen during civil wars when an outside power intervenes to take advantage of the situation. China has many examples. Mao and the Nationalists fought each other throughout the Japanese invasion, although sometimes they found it advantageous to work together. The Ming dynasty was fighting a civil war when the Manchus invaded in 1644 and fought both factions.

The Taiping Rebellion overlapped with several Qing-European conflicts. This created a situation where the Taiping Rebels were fighting the Qing in Central China and the British and French in Shanghai at the same time that the British and French were fighting the Qing on the seas, in Indochina, and around Beijing and Tianjin in northern China.