First of all, my personal belief is that the Kremean War within the context of DK's world is parallel to the real life Crimean War in terms of it's events and structure, and akin to World War 1 in terms of after effects.
SO, the real Crimean War goes like this: Russia (Obvious Kremling parallels) against allied forces (Bears, Kongs, possibly others), over control of territory (presumably the Northern Kremisphere).
But to understand the setup to the war, you have to consider the history of the region. "The Northern Kremisphere"… it has Kremling right in the name, as does Kremwood Forest, and Krematoa. My theory is that the Kremlings (in their imperialistic colonial stages some 200ish years ago) discovered the vast wilderness of "Bear Land" (as I'm going to call it for clarity, which will make more sense later). The Kremlings colonized it, leading to the naming of places such as "Kremwood Forest", and "Krematoa", among others.
This occupation had to have happened at some point in the past, but not as far back as the occupation of DK Island (as there are no Kremling ruins). However, the occupation ended at some point (presumably to Bear independence), and the Kremlings left but bitterly consider it rightfully theirs.
Fast forward to slightly before the Kremean War, and at this point, private Kremling companies start building mills and factories in Bear Land, much to the chagrin of the Bears (and the Kongs, who were anti-Kremling). Tensions rose, and the Kremlings began installing troops "for security of their economic interests", and eventually in a massive territory grab took over most of the place, starting the Kremean War.
The Bears (in alliance with the Kongs, which aided in troops and resources) ran a guerrilla warfare campaign, even taking to using the Kremlings pipelines to sneak troops around.
Also, I believe that "Krematoa" was the Kremling military's primary military base in the region, as well as their base of operations, which, either they sunk following the war to destroy evidence, or, a natural event caused its sinking (which possibly hurt their war effort).
Following the war, the allied forces put severe restrictions on the Kremlings economically and militarily, causing a decline in the empire and their overall quality of living. These conditions eventually lead to the rise of K. Rool's regime, and the invasion of DK island in DKC (see: Aftermath of World War I; Causes of World War II).
Here's some other odds and ends.
First of all, "Kremisphere", although a play on hemisphere, has nothing to do with a physical sphere, but rather the Kremling's sphere of influence. More specifically, "The Kremisphere" was initially a political term coined by the Kremlings to describe their own territory, and over the course of time has been used to describe "the territory they think they rightfully own even though they don't anymore". During the Kremean War, the allied forces took to calling the "Bear Land", as well as the surrounding seas as "The Northern Kremisphere", while Crocodile Isle itself constituted the Southern Kremisphere (although the Kremlings to this day consider all of DK Island and related islands to be within the "Southern Kremisphere"). Because "Bear Land" is the only physical landmass within the "Northern Kremisphere" region, the political boundary become synonymous with the land, and to this day the name has stuck as a hold out of wartime.
Next, some thoughts about K3. K3 (the snow world)'s name is a play on "K2", which is the world's second tallest peak (after Mt. Everest). However, K3 is clearly NOT the world's third highest peak. So, my theory is that K3 was a name given by the Kremlings (as with other places such as Kremwood Forest), and it's meaning is this: K3 is the third highest peak in the Kremisphere, ie, after DK Island and Crocodile Isle's peaks.
One other thing to note, is that the Crimean War is cited as being the first "modern war", which in terms of the Kremean war adds to its significance in DK's world. The Kremlings, Kongs, and Bears, have collided before, but the hostilities are in the distant past and are mostly fights for independence; for all we know, the Kremean War is the first major collision between the Kongs and Kremlings in hundreds of years, with the tools of modern warfare to aid them.