This is a loaded question, so I'll answer each part generally.

The Russo-Japanese War was fought at a time of imperialism when Russia was attempting to extend its influence into Manchuria and Korea (it was concerned with a railroad specifically) and Japan was attempting to get an imperialist foothold and establish street cred among the other Western imperialist nations. China and Southeast Asia, by this point, had been carved into spheres of influence by European nations and Japan was fearful of its own existence, so it decided to try to join the other western nations rather than wait for their inevitable demise. Thus, they decided they needed a foothold in Korea to have some territory cushioning Japan from outside influence. This brought Japan at a head with Russia, whose imperialist interests also lied in Korea, and China, who historically fostered a relationship with Korea subordinate to it. These issues were confronted first in the Sino-Japanese War of 1894-1895 and then the Russo-Japanese War, which ended in Japan winning Korea as a protectorate. Korea was declared a colony of Japan in 1910.

It stayed a colony until 1945, when Japan lost WWII and the empire crumbled. At that point Korea was destroyed because Japan's war policies were concerned only with Japan at the expense of territories. Basically, Korea (and all of Japan's other territories) were used just for their resources and its inhabitants were conscripted for labor. Thus, the United States and Soviet Union, the victors of the war, had the task of reconstructing Korea. They negotiated to split it at the 38th parallel (only because it was convenient since it was about halfway through and already drawn on maps). The plan was for US to reconstruct South Korea and USSR North Korea and eventually when they were fully rehabilitated, there would be an election for a ruler of the unified Korea and the partition would be lifted. For now, though, Korea would be split in order to account for the interests of both the US and USSR during reconstruction, which was a necessary point of negotiation as they had very disparate interests as we know from the ensuing Cold War. Germany was partitioned similarly. Also as we know, the plan to eventually unify Korea fell through. Kim il-Sung wanted to unify Korea under his rule, so he, backed by USSR, declared war on South Korea. The Korean War ensued, and it ended with an agreement reinforcing the 38th parallel, but it was a testament to the new geopolitical situation of the Cold War and a testament to the fact that Korea would now not be unified because each half was too important to the USSR and US.

The part about why North Korea is the way it is I know a lot less about, but I'll say what I know. It seems that the Juche ideology arose from Korea's recent history of foreign subordination. Korea was a Japanese colony for practically 40 years and was treated badly. North Korea was then subject to the Soviet Union's harsh occupation during reconstruction. The Juche ideology attempts to liberate North Korea from this history and become independent. Japan had Marxist influence for a time in the 1920s, but the military was fervently anti-Communist. The military which practically ruled Japan from 1931-1945 can be described as fascist. Of course Marxist ideology exists in North Korea today due to direct influence from Soviet occupation, but its government seems to resemble fascism more so. There's a strong sense of ethnic nationalism in the Juche ideology. Whichever way you describe it, ultraliberal or ultraconservative, it's still a totalitarian regime and that's where you can argue Japan influenced it.

Sources:

Japan: A Modern History by James L. McClain

Truman by David McCullough

Why did S.Korea thrive & prosper after the War yet N.Korea became very militant & authoritarian?

Well, the ROK was very militant and authoritarian right up through the 80's. However it was economically relatively liberal (although economic power was and is concentrated in a few Chaebol that had easy access to government capital), and it appealed more to industrialists who had succeeded under the Japanese. The DPRK was more focused on agrarian land reform.

Did Japan have any influences of North-Korean Ideology & the Juche ideals?

B.R. Myers claims that many of Japan's Korean propagandists found work in the North promoting a similar ideology, a race-based nationalism which portrays their own group as a pure race corrupted by foreigners.

Japan also hosts a lot of North Korean refugees who have provided economic and logistical support to the DPRK, and, for instance, Kang Chol-hwan came from a family of communist ethnic Koreans who moved to the DPRK and ended up in the Yodok prison camp.