This is a merger of notes from the curricular from my days studying history - a couple of years ago (mostly, Ole Georg Moseng, Erik Opsahl, Gunnar Pettersen og Erling Sandmo: Norsk historie 750-1537) and notes from relevant lectures. I don't remember exact pages from the curricular so I cannot give exact citations.

To put it bluntly, most people converted because they had to. Some earlier theories attributed the process to early missionary activity, but apperantly (according to my notes at least) there is consensus among historians that the process of converting Norway to christianity was initiated from the top down, with part violent, part peaceful means by regional leaders within scandinavia. Christianity meant outside controll over what had until then been private affairs. Family relations, and important social and political ties through friendships was of far less importance under a christian paradigm. Marriage and funerals was also regulated by the church, which had been a strictly private affair. Christianity and its church herralded a social revolution. That the process, at least in part, was a violent one therefore makes sense.

So, that answers part of your qeustion. Most people converted because they had to. But why was christianity "imported" in the first place? Sources indicate that Olav Trygvason was the one that started the process of converting Norway to Christianity (ca. 1000 a.d.). Apparently, the sources does not provide any personal motives Trygvason or Haraldson (Haraldson was a subsequent king, also quite set on converting people to Christianity) might have had converting people to christianity.

But! As players in a major tactical and political struggle across Scandinavia (and England) at the time, observing the Christian churches ability to organize the people under one single political institution, namely the kings throne, while at the same time providing economic and administrative support, it must have been tempting to try to emulate this in a country (Norway) that was notorious for it’s chaotic, complex and very unstable political alliances. Trygvason spent some time in England, which most likely inspired his later "crusades" in Norway.

When the church established a direct link between the one sitting on the throne, and god, the kings alliances was for the first time not dependent on his personal military resources. This is important. Remember, viking leaders personal resources often fluctuated, which in turn contributed to the regions unstable political climate. Now, the kings alliances and power was a tied to the throne, and whomever sat on it was annointed by god. His authority was (in theory!) unqestionable. But history does show that the kings position was only relatively more stable on the throne, than within the earlier alliance system.

An interesting set of subqestions arise from this. Why violence? Why not try to coax people into religion? This was most likely for tactical reasons. The existence of two religions, with significant overlap, would be a huge destabilizing factor within the region. The most important objective of the early Christian kings was to unite Norway under one christian king. Obviously, hedens posed a significant threat to this project. Therefore: convert, or be killed. Another aspect of this process is the way that christianity was introduced to people. It "borrowed" alot from pagan cultures in order to make for a more fluent transition. As /u/Searocksandtrees pointed out there is a whole lot on this topic here.

So, even though it is impossible to speculate on Trygvason personal motives, there was heaps of very good reasons to “import” Christianity from a tactical political and military perspective. I personally think it’s a bit of genuine christian missionary spirit and a bit of oppurtunism. People tend to adjust their world view to what corresponds with their (e.g.) political aims. Most people are not able to act in conflict with their core values, but they are able to adjust their values to circumstance to avoid this conflict. Cognitive dissonance and all that. So, in my mind Trygvason most likely was a genuine, full blooded christian, but became even more of a believer when he saw the tactical advantages within the christian institutions, when he would set upon uniting Norway.

Hope this was of help. Obviously english is not my first language, so excuse my spelling.

Edited for clarity!

I can't answer for the Vikings because it's not even close to my area of study but as for the Mongols there is a classic essay that seeks to answer that question by comparing the Khans with the Arab conquests: http://www.kobiljski.org/CUNY_GC_Spring_2010_Islamic_History/the%20conquests/Muhammad%20and%20Jenghiz%20Khan%20Compared.pdf

Basically what it comes down to is that the Mongol religion was not a universal creed and was largely concerned with legitimation of Mongol rule by an appeal to the divine. Mongol religion was in fact so unobtrusive in comparison with, say, Islam that there's a separate debate as to when it can even be said that the Mongols fully converted to Islam and gave up their faith.

Although its been a while since I've read it in detail the essay also talks about how political and military control remaining the domain of the Mongols exclusively meant that they could never hope to assimilate the existing civilizations they conquered by integrating the apparatuses of state.