Merge to HDR not Working in Lightroom Classic 10

Re. the suggested fix, I am not experiencing duplicate drives. I am experience duplicate drive names That is, the displayed contents on each of these "drives" is not the same.

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Also, after you've exposed the root folder on a drive, it should look like this:

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Yup, that screenshot confirms you've tripped over the duplicate-drive bug(s).   To fix that, for each drive, right-click one of the top-level folders and do Show Parent Folder; repeat that until you get to the root folder. Now you should be able to apply the recipe described in this article:

https://www.lightroomqueen.com/capitalization-catalog-error/

Even though in your case, both instances of the drive appear to have names with the same case, this method should work to get rid of one of the duplicates.  Be sure to make a backup (two backups) of your catalog before you start!

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The folders circled in red are not where they should be. The images/folders should be in the above folders labeled Lightroom 1, 2, 3, etc.  Also, I do not know how folders 2013-12-31 and 2014-12-7 got here. They were originally in one of the above master folders, and when I try to move them back to these master folders, I get a message telling me the folder is already there, but it is not shown there in the catalog. 

"It is even weirder than I thought. The files I mentioned above are in fact going to their correct folders, but the LR is showing them in completely separate folders, outside of any master folders."

Please post a screenshot of the Folders panel showing the right and wrong folders. LR infrequently does get confused about folder names, in a few different scenarios, and the screenshot will help identify if you fall into one of the known scenarios (and the known workarounds).

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"the last time I tried an HDR merge, I actually got an error message telling me the file would not show up in LR, which I had not been getting before. ... The HDR files are being included in the catalog, but they are not appearing next to the source images, in the same folder as those. They are going into their own separate folders"

Check if one of the photos being merged is in a different folder, in which case you'll get this message:

The merged photo will go into the folder of the most-selected photo.

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"The HDR file is clearly not showing up in LR, anywhere. ... Lightroom will not let me import the missing HDR files into LR."

You don't say how you searched for the file. Triple-check the file isn't in the catalog:

1. Restart LR.

2. Select All Photographs.

3. Do Photo > Stacking > Expand All Stacks.

4. In the Library Filter bar, search for the HDR's filename using Text: Filename Starts With filename:

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Try creating a fresh, empty catalog and merge from there.

I'm 99.99% sure that the images are not exported in a way that the export dialog is involved. I never add exported images to my catalog, and yet I can merge to HDR and/or panorama just fine.

Check your Export settings and make sure the box for Include in this catalog is checked. Export one, ANY, image to make sure that box is checked and stays checked.

Now you are going to ask why would that matter. IMO when you call up these feature the images included are actually Exported.