Origin, at least for games like ME2 which were released before Origin came out, literally just downloads a copy of the DVD, then runs the installer contained in that (same as if you were installing it from an actual disc). If you set it not to delete temporary files, you can probably even browse to the installer and run it yourself. So, since for most games you don't actually need Origin to be running, you can keep your use of it to a minimum. I haven't touched it since I reinstalled Mass Effect 2 (which was a real pain, but more because of the DRM than because of Origin).
It might work differently for games like ME3 and BF3 which were designed to use Origin from the start though, I suppose.
And Havok, didn't you say it was a DirectX internal error? That strongly implies it has nothing to do with Origin, it's more likely a bug in the game.