Nope, we had a war with you guys, we almost won but our president (Santa Anna or Saint Anne) got captured and had to sign a contract for his life. Texas was declared an independent country, one of the rules was: Texas will not join the US, they broke the rule, we got into war again (2nd Mexican-American war, or 2 second American intervention), we lost then California, Nevada, NM and Arizona, you could annex all Mexico but you dont wanted to 'cause racism (you needed to give all of us an american citizenship). You forced (later on) Santa Anna to sell La Mesilla to build your railway or you would invade our country again, he sold it.
Nope, we still say claim that status, you did it.
Yup, we also claim that status. New Spain (or Nueva España) was an authentic feudal colony in the XVIII century despite Medieval era ending and the Illustration stuff.
TLDR You stole the land, greetings.