Hamilton vs. Jefferson?

[True Federalist (진정한 연방 주의자)]

No, John Adams was always going to be the second President.  However, if Hamilton had not been forced to the sidelines, I think he would have stayed away from the shenanigans that led to Jefferson instead of Pinckney being our second vice-President.  The 1800 election would likely have been between Adams and Clinton, with Adams winning.  I think Adams would have followed Washington's precedent and not run for third term, and Hamilton would have been the most likely candidate for the Federalist nomination.  Hard to say who would have gotten the D-R nomination.  If Hamilton had won in 1804, it's quite likely we would have gotten involved in the Napoleonic Wars sooner than we did and on the side of Britain.  We probably would have invaded Louisiana in 1805 as our contribution to the War of the Third Coalition.  (I don't see a Louisiana or even a New Orleans Purchase happening with Adams as President.)

It would have been interesting to have a few U.S. ships at the Battle of Trafalgar, but I think the small U.S. Navy would have stayed mainly on this side of the Atlantic pond.

[True Federalist (진정한 연방 주의자)]

Hamilton was ineligible for election to the office of the Presidency.

Actually he was.  You need to either be a natural born Citizen, or a Citizen of the U.S. as of the adoption of the Constitution to be president.  While Hamilton does not meet the first possibility, he does the second as he was citizen of the United States as of 4 March 1789.

What I was thinking about, primarily, when I made this, was Hamilton's foreign policy given the trouble with the French. I'm wondering if this establishes an early precedent for American foreign intervention, and maybe even makes Conservatives warhawks earlier on than in OTL. Also, how America entering into war so early might effect it. How might these questions be changed with a Hamilton Presidency beginning in 1805?

Not nearly as much as they would have if he'd become President in 1797.  By 1805 it was clear that France was not going to be a bastion of republican virtue.  If Hamilton had tried to side with Britain in 1797, it could well have led to the breakup of the United States.  In 1805, war with France would have been a unifying factor both north and south.  The south would have had Louisiana and Cuba to gain from the war, while the north would have been able to maintain the more profitable trade with Britain.

Conversely, uninterrupted trade with the British will retard the development of Yankee manufactures.

[True Federalist (진정한 연방 주의자)]

Always hard to say what would be happen if you take people out of their own times into another.  Do you think Jefferson's wife Sally Hemings Jefferson would help or hurt him electorally?

Yes, I know he never could have married Sally Hemmings in his own time, but if transplanted 220 or so years into the future, I think he might well have. In any case, it's an example of how you can't neglect the effect of history on the man when you consider the effect of the man on history.