Okay, first those Terminal results you were asking for...
Last login: Wed Feb 15 14:49:23 on console
John-Neumanns-MacBook-Pro:~ John$ ps -cx
PID TTY TIME CMD
102 ?? 0:00.12 launchd
106 ?? 0:01.25 Dock
107 ?? 0:05.56 SystemUIServer
108 ?? 0:06.63 Finder
111 ?? 0:00.00 pboard
112 ?? 0:01.37 fontd
121 ?? 0:00.44 UserEventAgent
128 ?? 0:00.23 AirPort Base Station Agent
129 ?? 0:09.45 VBX6StatusItem
131 ?? 0:00.19 NetUpdateAgent
132 ?? 0:00.11 IntegoStatusItemHelper
135 ?? 0:02.29 CNQL3202_ButtonManager
138 ?? 0:00.05 iTunesHelper
139 ?? 0:00.18 Image Capture Extension
140 ?? 11:55.56 firefox-bin
151 ?? 0:01.54 mdworker
176 ?? 0:05.45 Preview
245 ?? 0:00.10 quicklookd
249 ?? 0:00.19 Terminal
251 ttys000 0:00.03 login
252 ttys000 0:00.01 -bash
255 ttys000 0:00.00 ps
John-Neumanns-MacBook-Pro:~ John$
Re internet ... yes, I overlooked the obvious. I have wireless in the house with a password ... don't know anything about encryption. Re Azureus - yep, I downloaded some items. I suspected that at first when the folder emptied (although I've never heard of malware that repeatedly erases a folder either) so I downloaded and ran VirusBarrier X6. It gave me a clean bill of health after scanning my entire hard drive, ie found nothing.
Re restarting. I can't recall if I restarted between the first and second disappearance. I do fully shut down every few days, but I can't recall if I did it exactly then.