Um… sort of, but not to the extent of today. It used to be quite typical for people to arrive on US soil without visa or passport up until around WWI or WWII. I don’t want to go over the entire history of US immigration, but suffice to say the rules do change every couple of decades.
No.
As noted, immigration is a Federal jurisdiction. The local and state level police do not get involved (at least, not outside Arizona) and in most cases have no desire to get involved. Anyone, regardless of immigration status, should have police protection if they need it. Having police involved in immigration law enforcement would make it even harder to police communities with large numbers of illegal immigrants.
Yes
It still happens.
Hell, there are illegal Mexicans in Canada, They didn’t just teleport there. There’s a continent-wide network of people smugglers and ways for illegals to get around.
Different levels of government in the US have different responsibilities. This is one sector of law enforcement attempting to usurp the domain of another sector.
It is especially problematic due to Arizona’s hypocritical past whining about background checks for firearms - you’re not allowed to pick and choose which laws you enforce.
It is especially problematic due to Arizona’s large Hispanic population - Arizona was held by Mexico before it was part of the US and thus the Spanish speakers were there before the English speakers. Their claim to the land is, in fact, of longer duration than the Anglos, but they are all too often treated as foreigners in their own home. This is a license for police to harass any Hispanic resident, legal or illegal, with constant requests to show papers or be deported. It has happened that America citizens have been deported by mistake so this fear is not entirely unfounded. There is considerable friction between Anglos and Hispanics in Arizona, with accusations of bias on both sides.
It creeps people out - the whole “show us your papers, please” meme seen in movies and television to depict Nazis and bad guy communists.
There is considerable sympathy for illegal Mexicans, to the point some believe immigrations laws are unfair and should be ignored. Some people believe there should be NO immigration laws and the border should be completely open. Some believe that, given the history of Arizona, Hispanics have more claim to it than Anglos. A certain number of Mexicans come to flee the drug wars south of the border and thus are refugees, which garners a lot of sympathy from a lot of Americans, given how many of the rest of us are descended from refugees and asylum seekers.
And lastly - immigration is deadly business in Arizona. People die every year attempting to cross the desert between Mexico and the US. Recently, a rancher on the boarder was murdered, it is believed by “coyotes” or human smugglers. Border patrol people have also been shot and killed on both sides of the border.
So, the legislation may seem innocuous to an outside but in context it is not.