For example, if President Trump invaded and claimed Sudbury, Ontario over a dairy dispute, could Alex Trebek run for President in the next election?
Maybe. It depends on the wording of the annexation treaty. Such a treaty would need to be written with explicit wording regarding the establishment of citizenship and eligibility of for holding the office of President of the United States of residents of the annexed territory.
If the treaty writers decide to stick with tradition, who knows. For territories annexed to date, the phrase in the treaties that established citizenship read:
the enjoyment of all these rights, advantages and immunities of citizens of the United States
Since there are no people currently alive who were born in an annexed territory, and no one from an annexed territory ever tried to run for president, there is no precedent that establishes whether this phrasing is sufficient to establish such residents as "natural born" citizens or not.
There have been several annexations throughout the history of the U.S. The annexation treaties contain clauses concerning the citizenship status of residents of annexed territories. For instance, the Louisiana Purchase stated it this way:
The inhabitants of the ceded territory shall be incorporated in the Union of the United States and admitted as soon as possible according to the principles of the federal Constitution to the enjoyment of all these rights, advantages and immunities of citizens of the United States, and in the mean time they shall be maintained and protected in the free enjoyment of their liberty, property and the Religion which they profess.
The Adams-Onus treaty says:
ARTICLE 6 The Inhabitants of the Territories which His Catholic Majesty cedes to the United States by this Treaty, shall be incorporated in the Union of the United States, as soon as may be consistent with the principle of the Federal Constitution, and admitted to the enjoyment of all the privileges, rights and immunities of the Citizens of the United States.
The Treaty of Guadalupe Hidalgo refers back to earlier treaties:
(Article iX): ... With respect to political rights, their condition shall be on an equality with that of the inhabitants of the other territories of the United States; and at least equally good as that of the inhabitants of Louisiana and the Floridas, when these provinces, by transfer from the French Republic and the Crown of Spain, became territories of the United States.
Article 2, Section 1 of the U.S. Constitution: "No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;"
This means that the person has to be born a U.S. Citizen; location is irrelevant. Alex Trebek, or any other Canadian-born person, would not be able to run for President if Canada were annexed, because they weren't born as U.S. Citizens.