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The United States paid Mexico $15,000,000 "in consideration of the extension acquired by the boundaries of the United States" (see Article XII of the treaty) and agreed to pay American citizens debts owed to them by the Mexican government (see Article XV). Other provisions included protection of property and civil rights of Mexican nationals living within the new boundaries of the United States (see Articles VIII and IX), the promise of the United States to police its boundaries (see Article XI), and compulsory arbitration of future disputes between the two countries (see Article XXI).

The phrase "extension acquired by the boundaries of the United States" means extension of its jurisdiction, not its ownership. The government does not 'own', it 'controls'. Its control is subject to the rights of citizens and inhabitants, otherwise there is no freedom.

Here's the actual Article XII from the treaty:

ARTICLE XII

In consideration of the extension acquired by the boundaries of the United States, as defined in the fifth article of the present treaty, the Government of the United States engages to pay to that of the Mexican Republic the sum of fifteen millions of dollars.

Immediately after the treaty shall have been duly ratified by the Government of the Mexican Republic, the sum of three millions of dollars shall be paid to the said Government by that of the United States, at the city of Mexico, in the gold or silver coin of Mexico The remaining twelve millions of dollars shall be paid at the same place, and in the same coin, in annual installments of three millions of dollars each, together with interest on the same at the rate of six per centum per annum. This interest shall begin to run upon the whole sum of twelve millions from the day of the ratification of the present treaty by--the Mexican Government, and the first of the installments shall be paid at the expiration of one year from the same day. Together with each annual installment, as it falls due, the whole interest accruing on such installment from the beginning shall also be paid.

Articles VIII and IX provide for protecting the 'property' and 'rights' of the inhabitants, ergo the Bundy family had grazing rights protected under the treaty. when Nevada was incorporated as a state, those grazing rights came under the state. The federal government cannot take them away unless by application of the 5th Amendment to the US Constitution.

Articles VIII and IX provide for protecting the 'property' and 'rights' of the inhabitants, ergo the Bundy family had grazing rights protected under the treaty.

You assume that the Bundy family was settled in the Bunkerville area prior to the Treaty of Guadalupe Hidalgo. You would be incorrect.

Mormons didn't settle the Bunkerville area until 1877, a full 29 years after the treaty. The Bundys, specifically, didn't purchase grazing rights until 1887 according to Cliven Bundy.

However, in 1887, the Bundys purchased grazing rights from the federal government not Nevada.