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For example, he used the Naboo crisis to be elected Supreme Chancellor. What would've happened if the Queen refused to call for a vote of no confidence or if Palpatine lost the election?

Then Palpatine is stopped or delayed. Frankly, given the amount of intrigue he's set in motion, if Padme doesn't follow his lead, he probably assassinates her, and uses that to fan public sentiment for his measures.

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Then, in Attack of the Clones, he manufactures a conflict between the Republic and the Separatists as an excuse to be granted emergency powers and legalize the creation of a military force. What does he do if he is never granted emergency powers? Does he serve out his term and go away or create some other, more serious conflict?

The conflict spread from the initial slapdown of the Trade Federation (causing other powerful groups to look askance at Republic interference).
Yes, if he fails to achieve what he sought, he continues to scheme in the background, and may never get his way. But in his case, he does.

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It would seem that he could've had someone else in the senate make the proposal but given that he relied so much on the absence of Padme and the manipulation of Jar Jar, it wouldn't appear so.

As the Trade Federation is a major part of the Separatists, that puts Naboo front and center for the emotional appeal. Naboo can state that the Separatists don't just want to be independent, they want to attack and control peaceful systems (as they did to Naboo).
Alderaan or Corellia would be arguing the matter over principle, and would need to be convinced that drastic measures are needed. Naboo is arguing from firsthand experience, and they are doing the convincing.

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Then, in Revenge of the Sith, when they Jedi find out that Palpatine is a Sith lord, they immediately go to his office to try to kill him. First off, is it a crime to be a Sith lord? Do they have evidence to link him to any actual crimes such as treason against the Republic? What if the Jedi had simply stood back and not done anything about Palpatine? Why not go to the Senate first and ask Palpatine to give up his emergency powers now that the war is over?

Well, given the Jedi's prominence in the Republic, being Sith may very well be tantamount to treason.
I believe in the EU, it was established that the Sith previously controlled the galaxy before being overthrown. I'm guessing it's something like finding out your elected leader is actually a Nazi or a Communist, seeking to revive Hitler's Reich or the Soviet Union.

Or to take a more medieval angle, imagine the Vatican finding out the French or Holy Roman Empire leader is actually a Satanist and a witch. Such an offense would not be addressed merely with words, they'd come to take his head.