If the folders have been imported, Lightroom will let you move and rename them.
The method sorting of the images within each folder has no affect on this.
Mac or Windows? What version?
What exact version of Lightroom (x.y.z)?
In great detail, what actions are you trying on the folders?
What error messages are you getting?
About 50 folders with image files sorted by file name in each. I trying to create groups of folders by creating new folders and then moving some existing folders into each. I had this type of file structure in Aperture and have done also this in Smugmug and Apple Photos
1) In Lightroom CC6, can you see subfolder contents in Navigator if subfolders and contents have been imported and organized by file name and not capture date?? Seems not. Could this also prevent moving folders into other folders, which I am not able to do with this file structure??
Yes but your images are Not "IN" LR. LR is a database program and only References, Creates a Record in the Database (IE the Catalog file), where you images are on your hard drives, internal or external.
I suggest you find where your images actually are, in which folders and on what hard drive that is connected to your system, and use the File Manager program for your operating system, Finder for Mac and Windows/File Explorer for Windows, and create and move whatever images you like into these subfolders. Then do a Synchronize in LR to Relink the images you have moved and folders you have created.
Unlike Apple's iPhoto, Aperture and Photos LR does not take control of your images. It allows you to chose what drive and folders your images are actually stored on/in (you can do that before they are imported or during the importing of them) and allow you to move them around whenever you like. But I suggest you do all the moving and creating of folders outside of LR and then Re-Link the images to the Record of them in the LR Database file, IE the Catalog.
In the Sort area there is an option to Arrange/Sort, That Is Arrange/Sort, your images inside the LR Library module by File Name. But that is only how they are Displayed in LR. It doesn't change where they are or how they are sorted on your drives and folders.
Your File Structure Is "YOUR" file structure. LR has nothing to do with that. LR Does Not have a File structure. It only Displays what YOU have created.
In Lightroom CC6, can you see subfolder contents in Navigator if subfolders and contents have been imported and organized by file name and not capture date?? Seems not.
Yes, you can have any folder structure you want. If you are getting Capture Date folders, you are most likely importing via the COPY option instead of the ADD. Then you won't get capture date folders.
Could this also prevent moving folders into other folders, which I am not able to do with this file structure??
Nothing is prevented. You can move folders to wherever you want.
In my opinion, you might want to consider a different approach. There are better tools in Lightroom than moving folders from here to there. You can obtain a more powerful and flexible organization of your photos by using keywords and other metadata to create organization and to search. And then the whole idea of moving folders from here to there is unnecessary.