I guess you do not use the same external HD at home. I mean, you could then just keep the library and thumbnails on that HD and simply plug it in before launching MC at either place.
Assuming you want to only occasionally add new files to the HD, but you would like to have your latest library data (playlists, views, tags, etc) always available, you can indeed use dropbox for automatically syncing the library.
One thing that you will need handle in a way or another is the drive letter. The media files (i.e. MP3, etc) must have the same directory paths. For example if the files at home come from C:\Music\... and the copy of this folder shows up as F:\Music\... when you use the external drive at work, you must fix that before the library can work at both places. Depending on how exactly your files are organized there may be a few ways to fix this.
I've created a clone library and saved it in my Dropbox folder. Should I now load the clone version?
If you want to avoid constantly loading the latest version manually, you can use the cloned version from now on at home and at work. At work you can simply create a new library that points to this "clone" library folder in dropbox and load it.
Then I'm a little lost on the thumbnail folder info posted.
The thumbnail cache contains small and fast copies of the cover art images (and images in general). The thumbnail views would not work at a useful speed if MC would need to read each and every image directly from the original source file when the views are browsed. All decent dedicated photo viewers use a similar technique.
By default the thumbnail cache location is inside the Windows user profile area (in "appdata"), but this can be changed by tweaking the "Thumbnails - Base Path" value. MC will then recreate the cache in the new location. For example, my dropbox folder is "D:\My Dropbox\" and I could set to thumbnail cache to be in "D:\My Dropbox\MC\Thumbs\". Of course, also the actual cover art files must be available at home and at work (e.g. "folder.jpg" in the album folder or embedded cover art in the file tag would work fine).
I attached a screenshot of regedit.exe and my actual thumbnail path on this PC.
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