No, it has not yet been asked.
Yes, you can install the viewer itself on an USB stick and put the chat and IM logs in a folder on the stick as well...
There is no way around this, but it's not really a problem, since you will still need to configure the viewer, even when ran from the USB drive, to match the computer's hardware it is ran onto (especially the graphics settings). You can then also configure the chat/IM logs folder ("Preferences" -> "IM & Logs" -> "Logging options" -> "Change path") while you are at it. Note that you will only need to do it once per computer (and since the Cool VL Viewer uses its own settings files, it won't interfere with any other viewer installed on the computer's hard disk).
Beware however about the cache folder:
do not put it onto an USB drive since it would quickly exhaust the flash memory write limits (this kind of memory's cells can be only written 1000 times or so before they degrade and stop working properly).
You may run any viewer in that way.