Hello everyone,
This is my first post to this forum, so I apologize if I have done so incorrectly. But I am writing in order to document my successful resolution of this problem! With great gratitude! I read and tried many different solutions, over the course of several months, and was finally told by Apple support that it was a hardware problem and that I likely needed to have my trackpad and keyboard replaced. But it never really made sense to me that my keyboard and trackpad could work for the login screen (but not after), and for it still to be a hardware problem. Not really wanting to pay $400 or so, I had finally resigned myself to using an external keyboard and wireless mouse.
Let me end the suspense and just say what finally worked: uninstalling Firestorm and my Second Life Viewer! I just did it on an intuition, remembering how often they made the computer freeze up. Now everything works fine again.
I hope that Apple (and perhaps Firestorm) will investigate how this particular bug could remain, despite my endless restores from Time Machine, trying every operating system from El Capitan back to Mavericks, wiping the hard disk clean and starting over [!], trying every sort of recovery mode, hardware diagnostic tests, clearing the PRAM, etc, etc, etc. Believe me, I tried every remotely connected suggestion, from every forum I could find.
I have a MacBook Air (13-inch, Mid-2013) currently running OS X El Capitan 10.11.1. And I feel like I just got my computer back.