Are surnames of non english origins anglicized in Usa?
some are, some aren't. Trump is Anglicized from Drumpf. Eisenhower was Anglicized from Eisenhauer.
No, not unless the person whose name it is wishes to
Many are. Mine was.
They can be if the person wants to do it.
It often happens
Some are, some aren't. It's the same in the UK. One of my ancestors' surnames was originally De Fougere - but after a few hundred years it had changed to Fugars.
Only if the person who has it Anglicizes it. When someone gets US citizenship, they are at that point allowed to make their name anything they want to. Some people Anglicize their names. Some people don't. Anglicizing names or even just picking Anglican names is most common for East Asians. Very often, someone's name who is like, Kim Lao-un, will pick a name like Joan Kim. Kim is their family name, so they'll leave that, but they'll very often just pick a name that they like that's Anglican. Nobody makes them do that. It's just something that many do. You might also find someone from, say, Spain whose name is Juan Martinez changing their name to John Martin, but that's a lot less common because of how common Latino names already are in the United States. Again, if anyone does that at the point that they're made a citizen, it's completely their prerogative. At the citizenship induction ceremony, people are asked what they want their name to be on their Naturalization Certificate, and working for Social Security, I've attended a lot of those ceremonies. In a normal courtroom induction of say, 50 people, you'll usually get one or two who change their name, and when they do, they Anglicize it. I've never seen it done where they've changed it to anything that wasn't obviously more Anglican. It used to be more common. So you'll find genealogies where the names changed when they came here. That used to be extremely common. When you go through records of Ellis Island, for example, you can find entire pages where people Anglicized their name because they didn't want to be seen as Irish or Polish or German or whatever. Speaking of German, after the World Wars, a lot of people of German decent changed their names to Anglican names. This wasn't the United States, but the British Royal Family changed its name and Anglicized it because of World War I, changing their family name from Saxe-Coburg and Gotha to Windsor, which was the name of one of the Royal Family's castles. Similar things happened in the United States because of so much anti-German sentiment because of so many American boys being killed by Germans. There's also a custom for historical names to be Anglicized. For example, the explorer credited with discovering the Americas was Italian and named Cristoforo Colombo, but he voyaged under a Spanish flag and so was called Cristóbal Colón, the Spanishized name, but in American history books, he's called Christopher Columbus, the Anglicized name. So if your question is regarding historical figures, know that sometimes they're Anglicized, but sometimes they're not, like Benito Mussolini's name is never Anglicized as Bennett Muslin, even though he's just as Italian as Christopher Columbus. Then there's famous people, like Jennifer Lopez. She's left her last name, but chooses an Anglican pronunciation for her name, because her first name in Spanish as said by any Puerto Rican and probably by tons of people she knows personally and within her family is pronounced "YEH-nee-fed," not like the Anglican pronunciation "Jennifer" that she's known by popularly. And, while again this isn't the US, when Prince William went to Chile for a year to build houses, he Spanishized his name, introducing himself as Guillermo, which demonstrates the custom that royals' names change to whatever it is wherever it's being pronounced, which is why we generally say the present monarch of Spain is King Philip and not King Felipe.
Over three (300) years ago they anglicized our surname to the English equivalent