In the hypothetical case that every member of the government (president and vice president included) wanted to call an early election, replacing the sitting president and vice president, would they be allowed to?
Short answer: No the constitution doesn't expect or allow for this.
But you ask for a hypothetical case in which (for some unspecified reason) Congress, the Vice President and the President wish to collude to effect an early election. So... with some interpretation of the constitution, and assuming everyone wants to play ball. The president and the vice president resigning in turn could allow the president to appoint any qualified person. If the Congress and the president want to hold an early election, the president could use this implicit power to appoint the president-elect
Congress instructs the states to appoint electors, who select a president elect
The Vice president resigns. The president can't fire the VP, but the VP can resign.
The President, with the consent of the senate, appoints the president elect as new Vice president.
The President resigns. The newly appointed VP becomes president.
The ability of a president to nominate a vice president can allow someone to become the president without being on the winning ticket of an election (This is roughly how President Ford came to power, despite never being elected: VP Agnew resigned. Ford was appointed, Nixon Resigned. Ford becomes president. Although of course Agnew and Nixon both resigned in disgrace after separate scandals - tax evasion and obstruction of justice respectively)
If the President and vice president could collude to nominate a successor for the remainder of their term, with the help of Congress they could, in effect, hold early elections.
Congress can decide the date of the presidential election. It would be possible for Congress to hold the election at any time, even in the middle of a president's term. Then following the Vice President resigns and for the President to appoint the president-elect as their VP, then the President resigns.
The effect is that the President-Elect has become president early, and gets to be President for the remainder of the term, plus the following term, for which they have already been elected.
Although this satisfies the letter of the constitution, it goes against its spirit, and of course this requires all parts of the government to collude. And there is no reason for a sitting president to do this.
If the president and vice president both want to replace themselves, they do not need the cooperation of Congress; they can just resign.
But there is no mechanism for calling elections to replace them in such a case; no matter how they leave office, the president would be replaced according to the well established line of succession, and the new president would appoint a vice president subject to confirmation by both houses of congress.
The only mechanism that exists for calling early elections is the mechanism of constitutional amendment as specified in Article Five. This requires agreement of at least three quarters of the states, so it perhaps does not qualify as an affirmative answer to your question.
In his book The Shadow Presidents, which he published in 1979, Michael Medved describes a situation that arose prior to the 1916 election, when the First World War was raging in Europe. In view of the contemporary international turmoil, President Woodrow Wilson thought that if he lost the election it would be better for his opponent to begin his administration straight away, instead of waiting through the lame duck period, which at that time had a duration of almost four months. President Wilson and his aides formed a plan to exploit the rule of succession so that his rival Charles Evans Hughes could take over the presidency as soon as the result of the election was clear. The plan was that Wilson would appoint Hughes to the post of Secretary of State. Wilson and his Vice President Thomas R. Marshall would then resign, and as the Secretary of State was at that time designated next in line of succession, Hughes would become president immediately. As it happened, President Wilson won re-election, so the plan was never put into action.
(I would put this in a comment but I can't because I haven't enough rep yet)
I don't think so, at least for the president and the vice president.
For congress (?) and senate (?), there are early elections (for example, see recent elections in Pennsylvania)
If both the president and the vice president are impeached at the same time, there are rules regarding who will succeed.
(edit), There is also a form of referendum (Recall Election ), but it is mostly at local and state level.
Not really. As per the US Constitution, US Presidential terms last 4 years. There is no provision for shortening or lengthening them. US politics is essentially governed by immutable cycles of 2, 4, and 6 years (House, President, and Senate respectively).
Technically, it is up to Congress to set the election date, so I suppose they could pass a law to have it happen earlier for an upcoming term, but that wouldn't change the length of the term, just the length of the "lame duck" period for the current term.
It might be instructive to look at what happened the only time there was a vacancy under the current rules (after the 25th Amendment changed succession in 1967).
During the 1973-1976 term both the VPOTUS and POTUS resigned in disgrace. Under the (current) succession rules, the new VP was selected by the POTUS, and confirmed by the Senate (much like is done with Supreme Court nominees). Due to that "confirmation" requirement, the Democratic Senate had a lot of power in the process, and reportedly informed the POTUS that the only Republican they would accept was the current House Minority leader, Gerald Ford.
Less than a year later the POTUS himself was forced to resign. Under constitutional rules, that made the sitting VP President. This meant yet another VPOTUS had to be nominated. Which is how the moderate Republican Nelson Rockefeller got to be VP.
In today's partisan environment, it would be easy to imagine a similar circumstance where an opposite-party Senate might refuse to consider any VPOTUS nominee at all, as the last Republican Congress did with the Merrick Garland nomination. In that case, were the POTUS also forced to resign (as Nixon was), the Speaker of the House becomes POTUS. That would probably be the current minority leader Nancy Pelosi (although she was challenged for the top leadership position in two of the last three Congresses, so it isn't a shoo-in).
As a very theoretical example, say the Democratic party takes over control of both houses of Congress in 2019 (highly unlikely on the Senate side, but we're being hypothetical here). The new Speaker would most likely be Democrat Nancy Pelosi. Lets further say they manage to use that control to either impeach or force to resign under threat of same both POTUS and VPOTUS (Mike Pence), without confirming any choice for VP in between the two. In this case the Democratic Speaker of the House would become President (until the 2020 election, the campaign for which will be well underway by the time all this political maneuvering reaches fruition).
No, because Early Elections are a feature of the Parliamentary System, and are called when the party in power loses a Vote of No Confidence. (There could be other reasons, but that's the usual reason).
The USA runs on the Presidential System with fixed elections. No matter how unpopular a President or Congressman is, he or she is there (barring impeachment, resignation, death, etc) for the whole term of office.