I'm a writer who will probably self-publish. I'm not a fiction author yet. I do not steal phrases from favorite authors, but I do steal phrases from all over, if they are generic but powerful. For example: the phrase "I just wanted my men safe," was in a war movie that I saw recently, and was powerful in that scene - it fits in a section of my non-war story and works well. I took it. The phrase "I very much doubt that you could do either" was similarly strong in another movie, but doesn't really apply to any of my story. So yes, ears open. SNL ex-actor Kristen Wiig says she keeps a little notebook to write down odd things she hears on planes, and uses those phrases, ideas, and quirks in her characters.
Certain ideas show up in my stories ... and I realize months later that they are similar to ideas from favorite authors. But, that's one reason why why those are my favorite authors.
I've seen some stories that are clearly driven in recognizable part from another story. Eragon looks amazingly similar to some of the stories of Pern. I recommend reading this link because Paolini has been accused of plagiarizing not just McCaffrey ... but also Tolkien and two other authors. If this is the sort of thing you have in mind - I strongly recommend against it.
I don't know the legal angle. There is discussion at the link above how that worked out for Paolini. There is plenty of ire directed his way (and even with Jeremy Irons, the movie stunk.)
Answer: It happens, there is a (legal) line to not cross, you will annoy some readers, and in the end develop your own voice.
As far as structure and flow, (not your specific question, but related) there are recognized formats that seem to withstand time at a mid- and gross level. Following Amadeus' advice, I diagrammed several paragraphs of favorite authors spanning decades, and will apply some of the structure (not words) to the beginning of my story. (I was surprised how similar the structures were between very different authors over forty years.)
More answer, to the title question: "Do writers copy other writers?" not "Should writers copy other writers?"
Yes. Writers copy other writers. There are caveats, such as the acceptable degree of copying, but that is not part of your question.
Here's a link.
Copywork was the primary way that schools in 18th and 19th century America taught children how to write. It was thought to be a highly effective way to teach students handwriting as well as proper grammar, punctuation, and syntax.
Yes. Writers copy other writers, which was your question.
Should you plagiarize? That's a different question entirely, and the answer is no.