A Shi`ee called mesbah asked these questions which are off topic and have nothing to do with the thread and Saqifah altogether, questions that have been refuted so many times while he brought nothing new to the mix.
What makes saqifa legitimate?
What makes any gathering legitimate, are you saying we're not allowed to gather and discuss matters of common concern with fellow Muslims?
A group of elites discussed a matter, the majority later agreed and obeyed their decision without any form of Fitnah, may Allah reward them.
Why everyone rushed to it? Why did they leave the body of their beloved Prophet (s)?
Not everyone rushed to it, only a few folks rushed to it, you can count them on the fingers of your hands. It's a good thing they rushed to it, otherwise a Fitnah would have taken place and the Muslims would have been split between those who gave Bay`ah to the Ansar and those who gave Bay`ah to the Mouhajiroun.
So may Allah reward them again.
Why some sahaba including Amir al-Mu'minin refused to give allegiance?
He did give allegiance, whether you want to believe before or after six month is your own issue but the end result is the same.
As for reasons he did not participate much in the first months, they are
Main reason: Banu Hashim were not consulted and they thought they had a right to it by means of nearness to Rasul-Allah (saw).
Secondary reason: The disagreement on the interpretation of the Prophet's (saw) words regarding inheritance.
Why the second caliph stoped Prophet (s) from writing his final will which would have saved the Umma from Misguidance?
His "will"? what "will" is this? No one said it was a "will".
As for WHY, Well because the Prophet (saw) was sick and in pain, so it is better if he does so when his condition is better after some rest.
Why some people refused to obey the Prophet (s) and join the army of Usama?
I thought the believers made sure that Usama's army was dispatched and they did what they were ordered.
Why the second caliph threatened people not to spread the news of the death of the Prophet (s)
Because he was in a state of shock and disbelief.
NOTE: I will not bother to go further than this, as I said do NOT attempt to change the topic at hand, that is really unprofessional.