I do not know how long you have had this issue, but first thing I would look at for the PC related issues would be the last time you did a Windows update ..
I had one that removed quite a few of my file associations, and caused other problems which forced me to rollback to a version prior to the update.
If you are copying the files to the same folder every time, check to see if that folder, or its contents have been blocked/locked
The way I read it, you are having problems copying the files from the LS to the thumb drives.
Are you able to see the files and other folders on the thumb drive using the LS? Make a new directory/folder? Export other types of files without issue?
If you have a small thumb drive, I would try formatting it, then copying the desired files to it to see if the problem persists.
If/When you succeed in getting the file to the hard drive of your PC, are you able to open and view the files with a text editor like WordPad, or Notepad++?
Granted the files are not truly easy reading, but at least with those editors you may be able to find something amiss.
Then save them with another name to see if they will open with your browser