Well, I only see two possible causes for duplicates.
Either you have MC's Auto Import set up so that it sees both the original location of the trimmed TS files (local HDD on PC) and the location on the UNRAID server, so Mc is simply importing the two separate file.
Or you have two Shares set up that point to the same location on the UNRAID server, so MC sees two separate paths to the TS files there, and hence considers them two different recordings, and so import both.
In either case, you should be easily able to check the path to the duplicate files within MC to confirm if that is the case.
I don't understand exactly what you are doing here:
What I want to do is to manually import new files by specifying the exact file in a browse. See it in the browse list, select it, and it imports just that file. I can do this with DVD and BR, but not with *.ts files. They won't show up in a import browse.
If you are using a View under te Video Navigator option on the left, then TS files won't show up within the MC "file browser" because they haven't been imported into the Library yet. That process may work for DVDs and Blu-rays if you ripped them in MC, or even just inserted them, as MC makes a record of all optical discs inserted in its "CD Database".
If you are you using Explorer under "Drives & Devices" to drill down to the DVD and Blu-ray files, then you should be able to find the TS files as well, as using this method doesn't require the files to be imported into MC first to be visible.
Of course, if you want to just import one file, and have the MC Windows Shell Integration Feature enabled, then you can just drill down to the TS files using Windows Explorer, right-click on it and select "Media Center > Import".
You are making it really hard on yourself though, wanting to browse, select a file, and then import into MC. Just set up Auto Import correctly, to only look at the location and for the file types that you want to import, and then let Auto Import find and import the new files for you. If you don't want to wait for Auto Import to launch automatically, just run it manually. If UNRAID shares File System Events (I don't think it does) then the import should happens as soon as the new file exists in a location that is watched by MC.
Have you read both of these carefully?
https://wiki.jriver.com/index.php/Importhttps://wiki.jriver.com/index.php/Auto-ImportSo basically, look very carefully at your Auto Import settings and UNRAID Share definitions, and you should be able to fix the issue permanently and make life easier for yourself.
Plus;
P.S. I did notice that sometimes the media server movie (file) name I name during editing trimming is different than the scrapped tag file from JRiver. Is that mis-spelling causing a problem?
The file name is completely different from the [Name] tag in MC. There is no mis-spelling. MC will only change the name of the file if you tell it to. Just check the filename in MC and you should see it is the same as the name you gave the edited file. The Sidecar file will be named appropriately to associate it with the movie file. i.e. For a movie file named "Star Wars Episode 3.mp4", the Sidecar file will be named "Star Wars Episode 3_mp4_JRSidecar.xml".