Did anything get transferred even though the FTP logs only show a connection, not any activity, and the two "public" folders were empty at the time? (The private folders were chokka with files)
Anyone with half a brain will do their utmost to cover their tracks. Trashing logs is what you learn at hacker nursery.... so you should assume your logs have been tampered with and are unreliable.
Has anything been silently installed on the (Flavour of Linux) NAS and should I therefore reset it to as-delivered defaults and start again?
Personally, I would assume yes. Much like a virus infested computer. You'll sleep much better if you do the proper thing and reset it (NOT using the reset button in the GUI, that would be a waste of time, but a proper reformat and re-install from CD).