So turns out our NSA spyware proxy may actually be Skype.
The intermittent http losses had been getting closer and closer together and it eventually got to the point where I lost it twice in a day. I was in the middle of burning Debian CDs (to reinstall the kids laptops, because apparently using the 7.1 cds I have and upgrading is not as good as downloading the 7.2 cds and using them) and quietly simmering about having to do a clean reinstall, when Josh came up with the brilliant idea to kill Skype because apparently Skype uses port 80.
I killed Skype and coincidentally just after it shut down http was working normally again.
Josh said everything should work fine now if I fired Skype back up again.
I restarted Skype and coincidentally lost http again.
Though this time when I killed it, http was verrrrryyyyyyyyyy sllllooooooooowwwwwwww so I ended up rebooting anyway after the cd had finished burning.
Josh had theories as to why it was using port 80 (getting around firewalls etc) but was puzzled as to why it would be putting a proxy between http and transport or something similar, and was even more puzzled when I checked Skype prefs and told him it was (theoretically) using port 8901 not 80. We then joked about how the port field was just a text box that didn't actually do anything.
I'm currently not running Skype to see if I encounter the problem again. In the meantime I was trying to move away anyway. I have a Jitsi account (which is an xmpp account), and fortunately not a lot of people to try to move. The hardest thing as per usual is critical mass and people not wanting to install yet another program or join yet another social network which I get.
The Skype I will probably keep around just in case I get another job with people that are ingrained in it.