I want to understand why people supported and the leaders justified their causes, not try to prove that communism is bad.
The Tsar wasn’t a good leader and it made sense that people were angry at him but what soon followed afterwards was the most progressive form of government which held democratic elections and gave more power to the people. Yes, there was a lot of fighting between the members but the DUMA which was a democracy is better than a violent gang of people that ban basic rights. The opposition which were the Mensheviks were the same thing as the communists since lennin caused a split in the party that supported a non violent revolution.
The Bolshevik’s even after assuming power didn’t give their promises and instead Lennin became the next Tsar. There was even a naval mutiny that happened against him and he created a secret police as well to eliminate free speech. The government was just as power hungry as the provisional government and there was a huge power struggle that led to a dictatorship. Nothing changed much after they assumed leadership and even worse things happened like how they lost more land to the Germans. Communists also believed that the workers will eventually win power over the rich so it seems stupid to cause an entire civil war that causes more death and suffering rather than stabilize your country first. It was more logical to accept the peace treaty with the Germans rather than give up on your promises and violate your own idea.
The Bolshevik’s also purged a lot of people unfairly despite their claim of equality. In the 2nd town the Tsar’s family was situated the house the engineer himself built had the builder kicked out and replaced with the family. How was that fair? The Tsar’s family wasn’t even that terrible and I would say that they were pretty good people for their time. The reason why they were hated was because Nick wasn’t suited to be a leader because of his lack of knowledge. Compared to other world leaders he was not that power hungry but naturally saw himself as a divine ruler like how the people saw him as. When they were still at their palace, the guards hated them but soon they started to like them and even helped them with gardening. That is a huge improvement from stealing Alexi’s toy gun. The room where they were executed was a room with a lot of bullet holes and bayonet markings proving that it was a brutal death where the children and the dog also died.
OK, there's kind of a lot here to respond to. I'm going to do my best to get to it all. If I miss anything, please let me know.
You have a few misconceptions I think that it's important first to dispense with.
First, the Provisional Government, which ruled from March to November 1917, although it intended to call for elections, never actually did so. To be fair, the World War was still going on, and once June came around, there was a ton of fighting in the streets and civil unrest that made it impossible. So while it's true that the Provisional Government was more (small d) democratic than the Tsar's government or the Bolsheviks', it wasn't formally democratic since it hadn't been elected.
Second, the Mensheviks were only one party in the Provisional Government; there were several more, most on the left but a couple in the center too (Octobrists and Kadets), so the dispute of the Bolsheviks with the Provisional Government was not a continuation of Lenin's dispute with the Mensheviks. It was much bigger than that. Further, while it's true that the Mensheviks opposed the Bolsheviks' seizure of power and that factions within the Mensheviks had emerged by the time of the civil war that were essentially democratic and against revolution generally, the Bolshevik-Menshevik split, dating back more than a decade, was more about Lenin's idea of the vanguard party leadership and its role in revolutionary activity. The Mensheviks saw this as essentially a form of elitism and thus rejected it, arguing for a broader, more democratic revolutionary leadership. The point is that multiple parties didn't participate in the Provisional Government -- mainly the right-wing parties but also the Bolsheviks themselves and the anarchists. (That naval mutiny you mention was an anarchist mutiny.)
Third, it wasn't power struggles that led to a dictatorship. Lenin led a dictatorship from the very beginning, and while it's possible that conditions might have loosened had he lived longer, he didn't, and so once the civil war was over and Lenin's health had all but disappeared, some economic controls were loosened, but the authoritarian form of the government did not change. What came later and did emerge from power struggles between Stalin and other factions was based on a previously existing dictatorship -- a point Lenin would have readily conceded. Everything that we commonly associate with dictatorship, like secret police, makes more sense once you understand that point.
Fourth, the loss of land to Germany was deliberate as the primary goal of the Bolsheviks once they took power was to end the war, and signing the Treaty of Brest-Litovsk, which ceded most of the Russian Empire's non-Russian European holdings to Germany, was how they got Russia out of the war.
That brings us to your specific questions.
Why did people support the Bolsheviks? I'd argue that there were two main, interrelated reasons. The first is where the Bolsheviks drew and built on their support, which was the cities. The primary constituency for a Marxist party was the urban proletariat, which was actually a minority of the working population. Nevertheless, around these workers, the Bolsheviks created workers' councils (called soviets in Russian) in which organizing could be undertaken -- a strategy known as "dual power." These councils drew two other populations: farmers leaving the countryside for the city and soldiers leaving the front -- the soldiers being the second key reason why the Bolsheviks had support, i.e., they intended to end the war. By the time the Bolsheviks overthrew the Provisional Government in November, they had built up a sufficient power base among the workers and soldiers in particular that they could at least seize the major cities and thus the government and then consolidate power from the top down.
Why do bad things if the workers are going to win anyway? Lenin's answer to this question, were it put, would have been that to assure a worldwide victory of the workers over capital, properly securing any previous wins would be absolutely essential. Not doing so would be like expecting to take Paris in World War II without securing Normandy first. Once power was established by the Bolsheviks, therefore, defending the worldwide revolution that they saw as starting with them was the top priority and it needed to be defended by any means necessary. If that meant disregarding the results of an election that you yourself have authorized, so be it. If that meant executing not just the Tsar but his whole family down to his dog, then he'll do that.
Still, did they have to do the really bad things? They'd argue yes, but you're looking for a more objective response, and I'm not sure one is available. No one is asking you to like the Bolsheviks, after all. But I do think it's important to have a thorough understanding of why they did what they did. From their standpoint, the Tsar stood for everything they sought to get rid of -- entrenched power, hereditary class, a rigid hierarchy, religion ensconced within state power, etc. As radical egalitarians, the best way to accomplish their goal was to get rid of the monarchy. The March Revolution had taken care of that, so what came next was a matter of keeping the Tsar and his political supporters from power and moving the ball downfield from there. This meant holding the government at all cost. The "real" revolultion, they believed, would happen in Germany, but they had established a key beach head and couldn't lose ground.
A final point I'd make is that it's easy to judge the Bolsheviks because their actions were often immoral from an quasi-objective standpoint. But that standpoint presupposes two things: first, that we could have done better under the circumstances, and second, that we are able to ignore the fact that we know how things worked out in the end. Neither of those things is necessarily true. Any government after the Tsar was on shaky footing at best -- the Provisional Government was nearly overthrown in July by an army general and only saved by the Bolsheviks having deployed its own loyal soldiers on the streets. And it's obviously impossible for you and I to just ignore how the Soviet experiment played out over time. The bottom line is that the Bolsheviks seized power and kept it and then they held it for 70 years. The Tsar couldn't hold power for more than a third of that time.
A recent book that I think covers all of this territory in a single volume and is highly readable is Laura Engelstein's Russia in Flames.
A very interesting primary source, albeit one too close to his material and thus highly biased, is John Reed's Ten Days That Shook the World about the Bolshevik seizure of power. It does give the viewpoint of a dyed-in-the-wool communist, albeit an American one.
Biased from the other direction but very good nevertheless is Robert Service's bio of Lenin, which also covers all this ground.
Finally, I'd recommend perhaps looking at what Lenin himself had to say. He was notoriously opportunistic and wont to change his mind as the circumstances required, but he wrote constantly and commented on current events, so if you want to understand why the Bolsheviks did what they did in the moment, he's a very good source of info.
https://www.marxists.org/archive/lenin/index.htm
If I missed anything, let me know.