I'll make a reasonably serious effort to respond to the invitation in the first posting.
1.The whole matter of languages would be different. I suspect you cannot get an education now such as Prof. Tolkien had. But language were a huge element in Tolkien's mental world, amounting to more than just inventing celeb to mean "silver" etc. There's much less emphasis on grammar in school instruction now, so the writer's imagination would not have had that experience. Those who have explored Tolkien's private papers on his invented languages will especially know what I am referring to. The modern writer might invent names, common nouns, etc. but these would not have the depth of Tolkien's. Tolkien found it natural to write poems in existing forms, use poetic diction, and to employ rhyme. It seems those now are regarded as old-fashioned and perhaps suspicious, "inauthentic."
2.Our modern writer would not know ancient and medieval literature in the original languages--not, anyway, to the degree that Tolkien did; and if he or she did know ancient and medieval languages, would not have explored them with the sense of original research that Tolkien experienced. That would affect the imagining of Middle-earth. Tolkien was a world-class scholar as regards Beowulf, the Saga of the Volsungs, etc. His imagination was piqued by the textual issues in these. Those are not such fresh topics now.
3.The writer would not be a World War I veteran. That had a huge impact on Tolkien's imagination, e.g. the Dead Marshes. Written today, we would get a long series of books in which the violent scenes were rendered in a much more close up manner. But Tolkien knew about real violence and suffering in a different way than the goremongers do.
4.The writer would not have grown up with the "naive" but deep patriotism that was part of Tolkien's culture. A modern writer grows up in a culture in which globalism is the default agenda of the pundits, politicians, Socially Conscious writers, etc. Is it conceivable that a modern writer would give us the Scouring of the Shire--which was completely amputated from the screen treatment? In Tolkien's original imaginative writings, at least, there were major connections between these ancient cities and locations in our own world.
5.A huge matter is that the writer would almost certainly not have grown up as a walker. Tolkien or one of his friends commented somewhere about how the private automobile etc. so greatly changed the experience of distance. People talk a lot now about valuing nature, and sometimes even take vacations in national parks and so on. They are not so likely to have daily rambles along rural paths, etc. These rambles naturally lent themselves to walking songs, etc. "The road goes ever on." But tell me when was the last time you heard someone walking along with hands in pocket whistling a tune.
6.Tolkien grew up not expecting government to do a lot for himself or anyone else. But now it's thought that government is there to "do things for" people. We as a culture obsess lately about "equality." Tolkien was more concerned about freedom. I don't think a modern writer is likely to have the trust that Tolkien evidently did have for the capacity of ordinary folk to look after themselves. His brother became a market gardener--sturdily independent so far as I know. A modern writer may hardly believe that such a thing is really possible, and his or her treatment of such an arrangement would probably be less heartfelt than Tolkien's.
7.In general people now are conditioned to "care about" a host of persons and things. These are often rather abstract. Tolkien knew the names of the wild flowers that grew where he lived. Who except hobbyists do, now? But that is part of his way of imagining.
8.Tolkien believed that reality is hierarchical. He believed that you explain a thing's being in terms of what is greater than itself (teleologically)--all the way up to Eru. A writer growing up in our time is taught to explain a thing in terms of what is less than itself, hence the ubiquity of the formulation "X is really nothing but Y." Our thoughts and emotions are really dependent on biology, biology really depends on chemistry, chemistry really depends on physics, i.e. the personal is less basic than the impersonal. That is very different from what Tolkien not just assumed for the sake of writing fantasy but really believed.
9.Feminism, modern social life, etc. have about killed the kind of exclusive, romantic love that Tolkien really believed in, experienced, and suggested in his writing. I don't suppose a modern writer would be likely to imagine Tolkien's Beren and Lύthien, yet that myth was one of the core elements of his imaginarium. True, a modern writer might have these two find each other: the "love of my life" thing. But the other would not have been the only love of their lives. Moderns' discomfort with Tolkien's imagination come to the surface in snide remarks about tapestry-weaving women, etc.
10.Tolkien believed in natural law or the doctrine of objective value (see his friend Lewis's Abolition of Man lectures at the University of Durham). He is very far indeed from our mainstream culture's scientism, which looks to "science" to give us the answers to what we should do (and which true science never can do, dealing as it does with observations of what is, not with oughts).
11.There are horrors in Tolkien's book, but the descriptions are generally brief and evocative, rather than detailed. For nearly 50 years horror fiction has emphasized "graphic" passages, and I suppose the Rings, written today, would do so.
12.A big difference that we can hardly conceive is that Tolkien's imagination was shaped by the written word plus the natural world. A modern writer's imagination will have been shaped to some extent, perhaps a very great extent, by television, movies, games, etc., in which the imaginary scene is done for the consumer and all the consumer has to do is let it wash over him or her. That is not how Tolkien experienced fantasy, and his reading of fantasy was in writers such as Morris and Haggard. For a discussion, see the J. R. R. Tolkien Encyclopedia ed. Michale Drout on 19th- and 20th-century literary influences on Tolkien.
13.Tolkien's imagination worked along the traditional, perennial lines of the loss of the golden age. But we've bet our children's, grandchildren's, and great-grandchildren's finances on a utopia that will justify colossal indebtedness, and losses of liberty and privacy, etc. Listen to the American candidates for president. How alike they are, in regard to being oriented towards Future Greatness. Gahhh. Our endless assembly-line of zombie movies, apocalypses (narrowly averted?), etc. suggest that at some level we don't really think the world they promise is coming our way, but we keep on electing them. I think there is a very great deal of hopelessness among people in our time. But hope is huge and heartfelt in Tolkien's imagination. It doesn't seem likely that someone writing Rings today would have that conviction.
As a thought experiment, try to imagine LotR as written by Mervyn Peake, a writer of great imaginative gifts, but whose sensibility was more modern than Tolkien's.
