Manifest destiny and slavery on acquired territories (specially Texas) motivated such conquest. See related questions.
Mexico was the most recent inquiry.
California
Northern Mexico, including the territories acquired from Mexico during the Texas Revolution and Mexican-American War.
California, Texas and parts of Arizona and New Mexico.
After the conquest, Mexico became known as Mexico.
NM
Any state that is not California, Nevada, Utah, Arizona, New Mexico, Texas, Wyoming, Colorado, Oklahoma or Kansas.
California
Enmity. Texas was viewed as a rebel territory, to be later re-acquired by peaceful means, but by force if necessary.
It was acquired from Spain. It never belonged to Mexico.
They didn't. After the Texas Revolution (1835-1836), Mexico viewed Texas as a rebel province to be re-acquired on a later date. After the Mexican-American War (1846-1848) Mexico was forced to stop any attempt on reclaiming Texas, as part of the Guadalupe-Hidalgo Treaty, which ended the war.