Can Bush suddenly change his mind and politely decline his nomination for re-election? What happens then? Does Cheney automatically become the Presidential nominee, does the Republican Party nominate someone else on the spot, or is there a special election?
Mind the forum, please. 
In theory, the convention would then reconvene and elect another candidate to run. Cheney would not automatically get the nomination.
Yes. There’s no law that requires him to run.
Has this ever actually happened?
George Washington declined to be nominated for a third term, thus establishing the tradition of the two-term president until F.D. Roosevelt broke it. That didn’t happen at a convention, though.
Oh, so this is just a tradition, not a law?
It was a fairly strong tradition. Several presidents probably could have been elected to a third term but declined for reasons of tradition. Then FDR broke the tradition, being elected to four consecutive terms as chief executive (dying in office early in his fourth term).
During Eisenhower’s term the Republicans pushed through a constitutional amendment limiting presidents to two terms in office. It was ratified without a hitch (and I think the majority of Americans are cool with it).
The Republicans briefly regretted the amendment during Reagan’s second term, wishing he could run again. Clinton restored their confidence that a two-term limit was the way to go 
The last person to decline a major party convention nomination was Governor Frank Lowden of Illinois, whom the Republicans nominated for Vice President in 1924. He declined immediately, while the convention was still in session, so the delegates took another vote and replaced him with Charles G. Dawes, also of Illinois. Party rules authorize the National Committee to fill vacancies arising after the convention, either from death or withdrawal or (in earlier days) a candidate being out of touch and not declining until after the convention was over. The Democratic National Committee last had to fill such a vacancy in 1972, when Thomas Eagleton withdrew from the ticket several weeks after the convention.
Actually that was during Truman’s presidency; it was ratified in 1951, IIRC.
And, contrary to popular opinion, it doesn’t actually limit a president to two terms in office, strictly speaking, although it has that effect. It limits a president to 10 years in office. This is why LBJ, while deciding not to run a second time, could have, even though it would have meant that he was serving in three different presidential terms. He served the last year of JFK’s presidential term. He served one of his own. He could have served another full term without exceeding the 10 year limit.
No, it DOES limit the president to two terms; it simply defines “term” so that acceeding to the office for less than two years does not count against the limit.
Amendment 22, section 1:
OK, related question…
If he accepts the nomination, but does not survive until the election (assasination, heart attack, something along those lines)*, what would happen?
The Republican National Committee would select a replacement. No presidential nominee has ever died or withdrawn between the convention and the election; if it ever happens, logic suggests that the Committee would make the VP candidate the new presidential nominee and choose a new VP candidate, but it wouldn’t be required.
If the death occurred close enough to the election, there might not be time to get the new candidate(s) listed on every state’s ballots, but since voters are voting for electors anyhow, this wouldn’t be an insurmountable problem.
If the candidate dies after the election but before the meeting of the Electoral College, then again the National Committee would choose a replacement for whom the electors would be expected to vote. In this case, logic would argue even more strongly for elevating the VP-elect. This has never happened to a winning candidate, but it happened once to the loser–Horace Greeley, who lost to U.S. Grant in 1872 and died three weeks after the election. Since he lost, it didn’t much matter who received his electoral votes, and the electors went in several different directions.
Once the electors cast their votes, the result is fixed; if the president-elect dies or refuses to serve, the VP-elect becomes President on Inauguration Day.
As long as we’re discussing hypotheticals…
What if both the (sitting) Prez and the VP die before the inauguaration but after the winning the election…
Of course the Speaker of the House would immediately become president… But what happens on Jan 20th? Does the Speaker stay president, or does whoever is Speaker at that time become President? (Giving the House of Representives defacto choice of * anyone * as President?)
The presidential succession act is not specific. It simply says the speaker of the House will assume the office of the President. That strongly implies that he will be entitled to whatever terms the former president was entitled to. So, if the president and the vice-president both die between the meeting of the electors in December and inauguration in Janurary, the person who was speaker when the president dies will become and remain president for the next four years.
Truman also made it very clear that he intended to run in '52 and that’s why the amendment reads:
*
But this article shall not apply to any person holding the office of President when this article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this article becomes operative from holding the office of President or acting as President during the remainder of such term. *
Else he (and the Democrats) would have hindered its passage.
In 1912 James Sherman, Taft’s running mate died after the election but before the electoral votes were counted. Nicholas Butler was picked by the Republican Party to receive the electoral votes for VP.
All eight of them.
Actually, Sunny Jim went south on October 30, just six days before the 1912 election. Taft informally designated Columbia University President Butler to replace him. By the time the RNC got around to ratifying Butler’s selection after the election, it was just a matter of who would get the eight electoral votes.