Interesting question and I'll try to go point by point, but first some general points:
a) There was no unified Armenian liberation movement in the Ottoman Empire. There were several ideologies and plans each practiced by different parties and even within parties there were noticeable differences between rivaling groups. Case in point: the "Dashnaktsutyun" or "Dashnak" party had the main plan: to secure basic rights for Armenians and get a semi-independent Armenia within the Ottoman Empire. Some small fractions advocated for a separate Armenia both from the Ottoman and the Russian Empires, a plan which naturally angered the Russians as well. And yet another group wanted the incorporation of all the Armenian lands into the Russian Empire, as it was the easiest way to stop the Armenian massacres.
b) the Ottoman Empire had offered Armenians basic protection from roaming Kurdish tribes and ensured their rights only on paper, and even that after much pressure from European powers and Russia. In practice, Armenians were second class citizens who were forbidden to carry weapons and hence were at the mercy of their Muslim neighbors. Furthermore, the Ottoman government not only couldn't offer any real protection to Armenians, as it had loose control over Kurdish tribes but as witnessed during the Hamidian massacres actively used them to suppress any dissent among Armenians.
Now to your questions:
- Because in 1920 Armenia had just gained its independence, had suffered Genocide and hence the country was awash with refugees, had territorial conflicts with both Georgia and Azerbaijan, was dealing with widespread hunger and disease outbreaks and had to deal with internal Muslim rebellions. Finally, in the autumn of 1920 Kemalist Turkey went to war with Armenia and won. The impending peace treaty had a very high price for Armenia, but at the same time, the Soviet army entered Armenia and basically saved it from Turkey. There was a rebellion against Soviet rule shortly thereafter and some regions in Armenia continued to resist them, but they were eventually crushed and defeated. But the main reason why there was no mass liberation movement against Soviet rule, is because the main mass of the population, the peasants, actually welcomed it, as they were offered rights and unlike in the Ottoman Empire were not living every day fearing attacks from nomadic Muslim tribes, lawless brigands or rampaging armies. In short, most Armenians had at last achieved physical security.
- Neither Armenians nor anyone else was second class citizens in the Soviet Union - that was basically the cornerstone of communism - there were no nations, everyone was brothers/sisters and allies and every people had the same rights. Armenia became a Soviet republic with its own capital which was incorporated into the Soviet Union with other republics (simplified sequence of events). The Russian Soviet republic with Moscow being the biggest and economically strongest naturally became the seat of power. But every nationality had the opportunity to rise through the ranks. For example, Joseph Stalin was Georgian and some Armenians, like Anastas Mikoyan (who was the second most powerful figure after Khrushchev) rose to very high ranks in the government.
- Many of them went into exile, like Garegin Njdeh and Dro (Drastamat Kanayan). Many of the political figures, especially members of the Dasnhak party, were arrested and killed. Armenia tried to be independent for two years, 1918-1920, but as I already outlined in the first part, it was hammered from all sides and the only viable solution was to fall under Soviet rule. Especially since in 1920 US Senate rejected to accept the mandate over Armenia and neighboring regions, as they did not want to send large armies to ensure peace in the region. And when Kemalist Turkey attacked Armenia, no European power came to Armenia's add, which had suffered genocide only a few years before. I think you can see why the Soviet army was seen as the only salvation for many Armenians.
- Although Christianity was important in the Armenian movements, it was not the deciding factor. The deciding factor was simply to achieve a situation when simple Armenian farmers would not have to wake up in the night to find some Muslims rampaging through their village, raping their wives and daughters, and putting to the sword the men. Yes, as Christians Armenians were not permitted to carry weapons, ride horses, or testify in courts against Muslims and paid much higher taxes, but it wasn't just Christians vs Muslims, it was in many ways also "settled people" vs "nomadic people". The main and very first grievance of Armenians towards the Ottoman sultan was the problem of the Kurdish tribes, which acted with impunity and in the middle of the 19th century started to take Armenian lands. Of course, European ideas and the proximity of Russia played a great role, but they were simply seen as the only viable solution to their problems: either to physically protect them or pressure the Sultan into doing something. The Sultan was mainly concerned with crushing Kurdish uprisings against the central government, but whenever the army was sent into Armenian populated regions, the rampages continued, only this time by the army. The problem was that at the beginning the Ottoman government simply did not have much motivation or power to change anything. Afterward, when European powers and Russia started to pressure the Sultan, the attitudes changed: now Armenians were considered a nuisance, an untrustworthy element which should be subdued by any means necessary.
So, to put succinctly, the main difference between living in the Ottoman Empire vs the Soviet Union, was that most Armenians had much better living standards, had physical security, did not live in a fear of a Genocide, were considered normal citizens and had the opportunity to enrich the Armenian culture. Sure, the suppression against Christianity was quite negative, but many people still remained believers, albeit without any external expression.
PS: I don't I remember any such Turkish plan (the one you refer to as "Erzurum plan"). Are you referring to Turkish plans for World War I? Could you eleborate?