If the U.S. adds a new territory, are the people currently living there able to become president? Or does the territory have to become a state in order for the people to be eligible to become president?
It can be complicated, and it usually comes down to the language Congress uses in the legislation that treats the addition.
Article II, § 1 of the U.S. Constitution lays out the eligibility requirements for a president.
No person except a natural born citizen . . . shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States.
To see whether someone is a "natural born citizen" there are a couple of places one can look.
First, 8 U.S.C. § 1401 deals with nationals and citizens of the United States at birth. It details a list of circumstances in which an individual may be considered a U.S. national by birth. Setting aside most of the situations that don't appear to pertain to your question:
The following shall be nationals and citizens of the United States at birth:
(a) a person born in the United States, and subject to the jurisdiction thereof;
The definition for "United States" comes from 8 U.S.C. § 1101, which deals with both nationality and immigration. Section 1101(a)(38) explains that:
The term “United States”, except as otherwise specifically herein provided, when used in a geographical sense, means the continental United States, Alaska, Hawaii, Puerto Rico, Guam, the Virgin Islands of the United States, and the Commonwealth of the Northern Mariana Islands.
Congress updates it frequently—for example to reflect the political union with the Commonwealth of the Northern Mariana Islands, which took effect on November 3, 1986.
But 8 U.S.C. § 1101 alone probably isn't outcome determinative. It's possible that one might still be considered a natural born citizen of the United States if other legislation (e.g., the covenant recognizing political union itself or a Congressional resolution) recognizes the status.
For example, people still debate whether Senator John McCain is a natural born citizen under 8 U.S.C. § 1401 or by virtue of other Congressional legislation. He was born at a military installation associated with the Panama Canal Zone in 1936, which was not widely considered a U.S. territory at the time. The next year (1937), Congress passed 8 U.S.C. § 1403, which retroactively awarded citizenship to those born in the zone on or after February 26, 1904. So some people argue that he became a natural born citizen retroactively under § 1403, while others argue he was a natural born citizen at the moment of his birth under § 1401.
For territorial additions (like the Marianas), presidential eligibility likely comes down to they way in which the legislation that recognizes the union treats citizenship. Births after the effective date of the union likely face a lower hurdle to natural born citizenship, while those born before the effective date may be more dependent on a retroactive grant.
This issue has never been addressed, even in dicta, by any court case of which I am aware, and I've read many of the relevant case laws in articles about this issue.
The language of the constitution is unclear on this point and its meaning in general is subject to a lot of debate by academics.
There are legitimate reasons for both interpretations.
Some of the analysis applying to foreign born children U.S. citizens (who are "natural born citizens), arguably points to a theory of the requirement that would hurt residents of the newly acquired territory because that case law looks to the status of the person on the day that they were born.
The original constitution overcame the issue for the new colonies by grandfathering in people who were in the colonies at the time that they became independent. This could be taken either as an implied precedent for newly acquired territory, or as a precedent for the fact that an exception for newly acquired territory requires express constitutional language, depending upon your approach to interpreting the text.