Were Mexicans welcomed into the United States

Yes, well they technically always were.

Explanation:

Mexicans have always been living in the southern portions of the US. No country just "lets" people into it and stay without that person getting a legal immigration status. Ever since the end of the Mexican-American War to the early 1920's, the US was very lenient on border control between Mexico and the US. Until the 1960's, Mexican Americans stayed at most a few hundred miles from the border. Only until more job opportunities opened up in the rest of the US the Mexicans started to disperse into the rest of the US. Even today, Mexicans can still apply for immigration status and get asylum. As long as you are legally allowed to stay in the US, you can't get deported.

No, the first mass deportation was between 1929-1936. There was about 400,000 to 2,000,000 mexican immigrants deported.