That's actually possible. But if you put a program on a USB drive, there would be a huge chance that it would be slow when used since it will get the files onto the drive and the speed won't be that good when compared to being stored on an HDD.
Re: Is a file moved to a USB drive recoverable in hard drive
To me, moving is just the same concept as deleting, only, the file is pasted elsewhere after deleting. You need to run the program if you want that file not being able to be recovered I guess.
You mentioned that this has started a week ago, have you made any modifications so far? Like may a security software installation or anything?
Try running a system restore, this is my favorite. Set to to one week back.