Still working on it. It appears one of the mods was either deliberately or accidentally 'damaged' causing these problems. The issue is that when I turn off all the mods system-wide, the problem is fixed. When I turn them all off individually, the problem remains. These, as far as I'm aware, are two different paths to the same goal.
We're going to be a bit wonky around here until I figure it out. No hacks whatsoever are going to be working, even though I have to wonder if it's something inside the plug-in system, that isn't a hack, that is the source of the headache.
Basically, something has been inserted into the site that makes it try to connect to 64.62.164.98, which is constantinople.hostgo.com, which is a relatively well documented spam site, a lot. By a lot, I mean, a lot a lot. That site is firewalled against us, which means we end up timing out on those attempts creating a black hole of PHP hell which sucks into a space and time fold that makes 7-8 minutes of nothing report as 60.5 second load times.
Moral of the story? I hate spammers almost as much as I hate computers. I can't live without the latter, but I can without the former -- so I propose that we legalize human hunting (only of approved targets of course), and we try to replace pit bulls with spammers and related vermin in the dog fighting culture. Dog fighting makes me sick -- but give a couple spammers shivs and a garbage can lid shield, and I'm all about watching that action.
