Is Puerto Rico domestic or international from the USA?

It's domestic, period, so no passport control. Never heard of CBP officials doing spot checks for flights to/from Puerto Rico but if they do, you could explain your Status to them in case you don't bring your passport. They will be able to check it out

Btw, once you get your Green card it's all you need even when entering the US from overseas. For example if you go to Europe, although a passport is needed for entry over there, once you get back to the US you simply present the Green Card.

Note: This answer was written specifically for green card holders and may not be entirely applicable to US citizens, visitors, or temporary residents.

Yes, Puerto Rico is part of the USA for immigration purposes, as is every US territory and possession. You can stay there as long as you wish. In fact, many green card holders live in Puerto Rico and other US territories.

From the USCIS document M-618 (rev. 09/15) Welcome to the United States A Guide for New Immigrants:

The United States now consists of 50 states; the District of Columbia (a special area that is the home of the federal government); the territories of Guam, American Samoa, and the U.S. Virgin Islands; and the Commonwealths of the Northern Mariana Islands and Puerto Rico.

There are a few other minor outlying possessions (such as some protected Pacific islands) which are also part of the USA but traveling to such places requires special permission. They aren't mentioned because travel to them is extremely rare, and in most cases no one at all lives there, but as they are part of the USA they still count.

Note that many US territories and possessions have different customs and agriculture/biosecurity rules than the mainland US, but this does not affect your immigration status. American Samoa, however, has different immigration rules and while you can visit you cannot reside there without permission.

Note also that you need to bring your green card if you travel to or from Puerto Rico or the US Virgin Islands. I also recommend that you (and everyone else) bring your passport. While this is a purely domestic flight, and it is not required for you to have your passport, if the flight has an emergency or there is bad weather at the destination it may need to divert to an airport at an island which is not part of the USA. It will be helpful to have your travel documents if this happens.