Editorial rejection: should I write back?

No doubt your first instinct is to challenge this desk rejection, but I would advise against it.

It is well known that appeals rarely succeed when referees go against you. When a handling editor thinks your work is not sufficiently interesting, I'd say an appeal is even less likely to be successful.

Coming specifically to Phys. Rev., your situation is somewhat worsened by the fact that alternatives to the Phys. Rev. A-X stable are not as attractive or popular.

But if your work is sufficiently useful, and helps advance your field, it will get cited even if you take your paper elsewhere. There is evidence for this in your case: You have senior researchers on arXiv giving you good feedback.

After two weeks, I got an email from the editor saying that it is being rejected without any review. My friend has got such rejection from the same journal, he received exact same text, which means that it was simply copy pasted.

This is very likely true - journals get a substantial number of submissions, and for most actually tailoring their desk rejection emails would be overly burdensome, and likely not terribly productive.

Now, I am a researcher who has a small number of citations, I have never published in Physical Review before and I am from a third world country. My feeling is that these things have biased the editor against the manuscript even though it is a good work if not groundbreaking and he should have at least sent it to reviewers.

While it is possible that those things are true, you have zero concrete evidence that they are. Most papers submitted to a journal get desk rejected. There are myriad reasons for that - it's possible the flaws are stylistic, rather than rooted in the results. It may not be a good fit. There may be more compelling items in the queue. The editor may simply not share the opinion of the people who have emailed you.

All of those things are possibilities, and far more likely than the editor being biased against the paper due to who you are.

As for whether or not you should email and appeal that decision, I wouldn't bother, as I'd place the odds of success at indistinguishable from zero. While I often encourage colleagues and students to appeal editorial decisions if they have specific ways to address a review that they believe is in error, etc., merely "I think it's good enough" isn't enough fuel to do that. You'd be better served prepping the paper for another venue.