Solution:
Apparently there was an additional keyboard attached to this computer, and it has media keys, one of which was getting pressed just enough that desktop vibration would make the contact.
So if this is happening to you, remove all your USB devices just in case some joker attached a USB hub or long uSB cable and hid a pranked keyboard under your desk.
I ended up using Process Monitor and looking at the log immediately prior to the launch of the default browser. I noticed that Explorer.exe was looking up keys in the registry that have to do with buttons on multimedia keyboards (for instance you might have a keyboard with an internet button).
So a key similar to HKLM/software/microsoft/windows/currenversion/explorer/appkey/7
will usually have "association
" set to "http
". Setting that to "" disables the internet multimedia key.
I don't have a keyboard with that key, but I made the change anyway, as it seems that some process is sending that keypress to the system.
The random windows are no longer popping up, though I'm going to let the system run overnight just in case.
Still means I haven't found the root cause (who is sending that keypress?) but I've found a workaround.