Your framebuffer looks to be the generic one (terminal says ATY,RadeonFrameBuffer, which is the generic one), meaning you probably aren't getting full QE/CI (is DVD Player crashing?). You aren't using Chutoro, so your AtiConfig line isn't doing anything. On the bright side, it means you could hack together a driver to make it work if none of the default framebuffers do it for you.
What model exactly is your 7770? As in, which sapphire ghz edition (since they have a few variations)?
The HD 4000 only shows up at all under OS X if it is set specifically as your primary video device in the UEFI/BIOS, and is also set to enabled (on non-gigabyte boards) or disabled (on gigabyte boards).
I'm not sure about the artifacts, but what is the device ID and vendor ID of your card?
I'd like to add it to the list of working cards in the second link of my signature, if that's alright by you.