1. According to Avos the machlokes of Korach exemplifies machlokes she'lo l'shem shamayim. Isn't that a strange statement? The whole problem with Korach's machlokes is that it was she'lo lishma?! That's like telling a guy who drives to the Reform temple to daven on Shabbos morning that his tefilah is lacking because he doesn't have proper kavanah. That's the least of the problems! Here Korach undermined the authority of Moshe and brought the whole transmission of Torah into question and our problem is whether he said a l'shem yichud and did it lishma or not?!
If indeed Korach's argument had even some theoretical validity and it's only the she'lo lishma that's the problem, why is that such a big deal? Mi'toch she'lo lishma ba lishma! Sefas Emes writes that you see from here that machlokes is an exception to the rule. A machlokes can be constructive -- halacha is enriched by the debates and machlokes throughout sha"s -- but that's only a machlokes undertaken purely l'shem shamayim for the sake of Torah. A machlokes with an ulterior motive results only in destruction.
Maybe the key to the Mishna is a Netziv in P' Acharei Mos that we discussed here. Netziv posits that even though a person should learn Torah even she'lo lishma, one cannot be mechadesh in Torah and innovate new practices unless one's motives are pure lishma. Had Korach expressed his innovative ideas, "ki kol ha'eidah kulam kedoshim," purely l'shem shamayim, perhaps he would have gone down in history as the intellectual foil to Moshe, the Shamai of his generation, a view that was rejected, but a view that had validity. However, since Korach acted she'lo lishma, his views by definition were no longer a chiddush, but rather were a distortion of Torah.
2) According to Ibn Ezra the story of Korach's rebellion really took place much earlier, when the Levi'im were selected to take on the role formally assigned to bechorim. The bechorim were unhappy with losing their jobs; the Levi'im were unhappy at being assigned to role of helpers to kohanim; the stage was set for rebellion. Ramban disagrees, as, in general, Ramban takes a far more conservative position when it comes to re-ordering parshiyos. According to Ramban the parshiyos are written in chronological order and the rebellion occurred after after the episode of the spies, after the people were reeling from having heard that the dream of entering Eretz Yisrael would not come to fruition during their lifetime. Korach was able to use that disillusionment to his advantage in pressing long held grievances of his own.
What is Rashi's view on this issue? On the one hand, Rashi at the beginning of Korach cites the Midrash that says that Korach mockingly asked Moshe whether a talis completely dyed with techeiles needs a techeilis string of tzitzis -- the Midrash sees the Korach story as following the parsha of tzitzis, which itself is part of the response to the story of the spies. This seems to accord with Ramban's view. On the other hand, Rashi in the beginning of Shlach asks why the Torah juxtaposes the story of the spies with the story of Miriam's lashon ha'ra. The implication of Rashi's question is that these stories are not written in chronological order, otherwise chronology itself would be a valid reason for the juxtaposition. What is the out of order parsha that should chronologically appear between the Miriam story and that of the spies? Mizrachi suggests that perhaps the answer is Korach's rebellion. (Mizrachi does not see a need to reconcile these two Rashis. He argues that Rashi need not be consistent; Rashi can adopt one Midrashic view in one place and a different view elsewhere depending on the local textual problem he is trying to address. See Maharal in Gur Aryeh who disagrees.)
According to this latter view, we don't have the disillusionment that followed the episode of the spies as a motivating factor in the success of the rebellion; we don't have the immediate role changes of Levi'im and bechorim that Ibn Ezra focused on. What we do have is the parsha of Eldad and Meided prophesizing that Moshe would die before completing his mission. What we do have is Moshe himself saying that were it only the case that everyone could be a navi. What we do have is Miriam equating to some degree Moshe with herself and Aharon and not recognizing his uniqueness. Perhaps that context explans why the call of "ki kol ha'iedah kulam kedoshim" took on greater meaning. If everyone could be a prophet, if all prophets were to some degree the same, then why not spread authority to all?