Halavista's solution worked perfectly, thanks!!!
Just for the record,
Here was the problem:
I had 2 USB flash disks (both FAT32 formatted) and also tried a friend's windows USB disk, which would not mount properly (seemingly only after OS X 10.9 "Mavericks" update). They'd show up in the sidebar & in my disks list (and in Disk Utility), but if I tried to open them in Finder, the window would briefly flash something like contents but go back to the previous screen - I could not view/edit the contents of the disk.
Disk Utility could Unmount & Mount the disks, but that wouldn't alleviate the problem (and would fail upon Varify/Repair with "could not unmount."). The disk's root folder would also show up in Terminal /Volumes/, but attempts to copy files via terminal were met with strange results such as "The file xzyabc already exists.", with the file having Not been copied when viewed on another computer. Various methods of Reformatting (always FAT32, since I was transferring to Windoze) also did not solve the problem.
As halavista said, this solved my problem:
I followed the instructions here to reset the System Management Controller (SMC):
http://support.apple.com/kb/HT3964
Note that the only indication that the SMC reset occurred is given in this hint:
https://discussions.apple.com/thread/4615991 -- the magsafe power adapter LED may blink. I'm not sure if mine did, I wasn't watching it at the time.
After booting back up, both of my USB flash drives are acting normally - I can read & write them, eject/re-mount and see the new contents.
Maybe the Apple SMC article should list this USB drive issue as another reason to reset the SMC.
(Aside; The SMC article says that this would also solve the fans spinning up to full blast - which my computer started doing recently. The SMC reset does Not appear to have stopped that.)
My specs: MacBook Pro 2.8 GhZ Intel Core i7 // Mac OS X 10.9.2