Well, back in the day when I used to do this:
How do you plan, research, write, etc.?
I'd user internal thinking to build up a model. Then I'd check it for consistency a whole lot, and when satisfied, use intuition to build a narrative. This narrative is often a 'first draft', but is usually a random stream of outpouring that would never be a proper first draft to show someone. I would then take that draft and 'sit on it' for a few weeks. Then I would re-read it and translate it into what I called "readable human form". This would be the actual first draft. One could almost think of the former document as an 'externalizing narrative in note form.'
How do you approach research and paper structure?
As someone who likes to read, I was always reading and adding things into an internal taxonomy. This taxonomy was also recorded externally in paper folders, and in computer form via a giant tree of recursive directories (folders) with subtopic folders inside topic folders. Then when it came time to cite sources and/or reread, I would descend the topic tree to find the point I was looking for. I'd say that organizing topics and subtopics, and making bridges between them was my strong point. My actual writing was far weaker.
As far as paper structure goes, my natural narrative style is very difficult to read. I relied on a rather artificial style that divided a paper into many sections and subsections. This was not out of choice. It was a necessity. A natural writing talent need not use this approach.
My argument structure was generally the typical conference/journal paper structure, with a problem statement up front, a direct statement of the conclusions I would make, and then the evidence I would present. Some professors hated this approach, claiming it was redundant, but I prefer a paper does this up front so that I can check them on it as I read the rest of the paper. I hated reviewing papers for a journal that implied one set of claims up front and then presented another in the body of the paper. Instant rejection in that case.
How long does it take you on average to write a major research paper?
In my case, some papers were written in a month. Others evolved over a year. This slow pace is one of the reasons I left academics. A good academic should be able to turn out a high quality paper with 3 to 5 days of 24-hour cycle working. (After spending months doing research.)
Do you write it all at once or in parts? Do you write at the last minute?
I like to write it all at once, and then mutate it draft after draft. Sometimes sections would change, or the narrative order would change. Occasionally a paper would go through periodic shifts in form (particularly the weaker ones.) My papers were definitely not written at the last minute. They were organic things that grew over long periods of time. Many papers I contributed to in groups WERE written rapidly. But the rapid process was more the preference of my colleagues and mentors, and better suited to experimental papers with obvious conclusions.