I had to pare down the original question. My hope is that you repost it as separate questions because there is so much in the original worthy of exploration and thought.
We can presume that Karl Marx would feel that the election of the President does not reflect the inner soul of the people. Marx would probably argue that the reality of dialectical materialism orders everything, especially political rule. Marx's view of political rule under a capitalist government is one in which government consists of the wealthy, for the wealthy, and by the wealthy. The election is nothing more than a contest between individuals that represent the interests of the bourgeoisie. For Marx, since economics is the ordering principle to all human interaction, the election of the President is not going to feature candidates that have the proletariat's interests at heart. Rather, the two party system has done a good enough job ensuring that the wealthy will remain in power no matter which one of their candidates is elected. Due to this, elections do not represent the true or inner soul of the people. Rather, it represents the entrenched forces of wealth that continue to surrender power to the majority of the body politic, the proletariat.
Marx would continue to point out that the Electoral College as the agent of electoral action demonstrates this. Conceivably, even if the will of the people is heard in electing a candidate that represents the essence of the proletariat and wishes to make structural changes to benefit the proletariat, then the Electoral College could stop this from happening. It stands to reason that the popular vote of the candidate who represents the proletariat's wishes would fly in the face of the establishment, embodied by the Electoral College. In any case, Marx would feel that the presence of dialectical materialism precludes any real election representing the inner soul of the people.