Why is the Vice President elected in the US? Why isn't he chosen by the President like any other secretary? Is there any important reason why the writers of the Constitution made that choice?
How is the Vice President elected in the actual bi-party system? Is possible to have a VP from party B even if the President is from party A?
The original conception of the US presidency was very different than it is today. Originally, there was a pool of potential candidates selected, out of whom the electoral college members would vote for their two favorite candidates. The one with the most votes would be the most qualified, thus he would be the president, and the one with the second most votes the second most qualified, and thus the Vice President.
The second president, John Adams, was elected, and his arch rival (at least, politically speaking), Thomas Jefferson, was the vice president. This didn't work very well, and in fact, in later days, the president asked the electors to vote for a second candidate alongside him, such that the two had similar views. Well, that also didn't work, leading to the 12th amendment.
So, what does this all mean? For all practical purposes, the vice president is a historical artefact. Still, it adds another dimension to the presidency, and allows the voters to select not only the president, but the person who will fill his role if something should happen to the president. By the succession laws, the next person in line is the Speaker of the House, which is a semi-elected position (elected by elected officials from elected officials, plus being elected from their district), as is the next person, the President pro Tempore of the Senate. These two people have other responsibilities; namely, to lead the congress. Beyond them is the Cabinet. Personally, I would prefer to vote for them as well, but there needs to be some freedom in allowing the president to change these people around.
The vice president has no official responsibilities in government, aside from casting a vote to break a tie in the senate. This person is literally a fall-back president. Thus, the people of the United States prefer to elect this person, and allow the President the flexibility to assign his Cabinet, who do "real" work. Note that these still must be approved by Congress, giving them a semi-elected role as well.
The Vice President is elected because it is constitutionally required, as per the 12th amendment. The reason the Vice President is elected is because, since the VP is first in the line of succession, and that is one of the VP's primary responsibilities, so it was seen as a position that should be elected. Alexander Hamilton outlines this view in Federalist No. 68:
The other consideration is, that as the Vice-President may occasionally become a substitute for the President, in the supreme executive magistracy, all the reasons which recommend the mode of election prescribed for the one, apply with great if not with equal force to the manner of appointing the other.
As for how, the electors in the electoral college cast a ballot for the president and vice president. Both votes are technically separate, though in practice Presidents run with a Vice President.
As for the rest of your second question, before the 12th amendment, the runner up in the election was made vice president, so it actually used to be the norm that the two would be from a different party. Today, it would only happen in the case of an electoral collage tie, in which case The House of Representatives selects the president, and the Senate selects a VP.
The Vice-President IS chosen by the President. The Vice-President "gets in" if and only the President gets in. Put another way, the President and Vice-President now come in a "package."
Because the Vice-Presidential candidate is elected in this way, rather than being appointed after the fact, the selection of the Vice-President is seen as another "plank" in the "platform" of how the President plans to govern. As such, it's another way for the President to define his candidacy and win (or lose) votes.
Your alternative 2, which included the possibility of the President and Vice-President from different parties was the way it USED to work, under the original Constitution, before the 12th Amendment. The two men most affected were John Adams and Thomas Jefferson, two ideological opposites, that had trouble working together. The 12th Amendment was passed in 1804, late in Jefferson's first term, so that Presidents and Vice-Presidents AFTER Jefferson wouldn't go through what he and John Adams did.