Finally, a question I can answer!

First and foremost, there was nothing keeping Hamilton from running, except his political apathy (and death by gunshot wound). Nobody was a natural born citizen at that point, they were all British subjects, and Hamilton, while born in the Caribbean (Barbosa i believe) was still a British subject as it was a British territory. It's similar to how McCain was able to run for president in 2008 despite being born in Panama.

I have no sources as to whether or not Hamilton expressed any desire to run for president. However, there was discourse among the federalists as to which of the party leaders (Adams or Hamilton) should represent them in the 1800 election. Ultimately they felt it unwise to run Hamilton, he was a very controversial figure.

Now as to the creation of the continuation:

It's difficult to say because, as delegates were prohibited from taking notes on the goings on at the constitutional convention in Philadelphia (that of course didn't stop James Madison) so they could speak freely without fearing that it would tarnish their reputation.

Alexander Hamilton was preset in the constitutional convention, he was the rep from NY, so I can't imagine that they would present it as such while he's in the room, but even if they did, it still doesn't keep him from running.

Edit: Hamilton was born on the Island of Nevis, not Barbados as I believed (or Barbosa according to iPhone Autocorrect.)