1. He never got a good chance to be president. In 1796, John Adams was heir apparent -- Adams felt he deserved the presidency after 8 years suffering through the vice-presidency, and his stature was such that nobody within the Federalist Party was going to seriously challenge him. Granted, Adams certainly didn't win by a large margin, but there was no way Hamilton could have made a serious bid for the presidency. And by the time the next presidential election rolled around in 1801, Hamilton's political career was on the descent. His hawkish stance towards France, his insistence that the U.S. cultivate a standing army (of which he would be general!) and his behind-the-scenes manipulation of Adams' cabinet members (who were loyal to Hamilton, not Adams) made him bitter enemies with Adams and didn't win him friends with more moderate Federalists.

And then there's the question of whether or not he really wanted to be president; as you mention, he was content to stay behind the scenes and manipulate from afar. He did this quite well. And he knew that no matter what he did, his background was always an easy thing to throw back in his face, an easy way to discredit his argument. Finally, had he not died, would he have run for president later? (assuming his political career came back?) He might've entered public office again -- he wasn't really the type of guy to just hang out and let things pass him by -- but I doubt he would've gone for president.

2. So the sex scandal came out because of a corruption scandal -- Hamilton confessed to the charges of adultery to absolve himself from possible charges of corruption. James Reynolds, husband of Maria Reynolds, had blackmailed Hamilton for years, and some influential Republicans saw those transactions and thought they were evidence of embezzlement. When three of them came by came by (I can't remember their names off the top of my head, but one was James Monroe) with accusations, Hamilton explained the whole sorry adulterous affair, which totally made its way back to Jefferson and definitely did not stay within the confidences of the four men. As gossip started to spread, Hamilton later felt the need to explain this in a public forum; he did so by publishing this lovely, lurid pamphlet that went into all sorts of detail about the affair, etc. This was the sex scandal part of it.

As for how this impacted his political career? If you didn't like Hamilton before the scandal, you definitely didn't like him after. But his allies still stuck with him, for the most part; Martha Washington even sent Hamilton's wife Betsey a pair of slippers as a sign of support.

3. This one I don't know, unfortunately.

For further reading, Joanne Freeman's Affairs of Honor is a really fascinating look at the politics of the 1790s and what it meant to be an "honorable" politician. This idea of "honor" really shaped how the founding fathers navigated their political lives, and was especially relevant to Hamilton, who cared about honor a great deal. (Freeman goes into a lot of detail on dueling as a performance of honorable men, and Hamilton was involved in a lot of duels -- not just the one that killed him, but in one with James Monroe over, you guessed it, the sex scandal).

Hamilton was killed in a duel at the age of 49. Washington, Jefferson, Madison, and Monroe all were elected president in their late 50's. Adams was in his 60's. So Hamilton might have run had he lived longer, but probably not. Support for the Federalist Party was waning by then.