I'm on a Radeon HD5870 and seem to have solved my problem with the DirectX crashing before cutscenes (knock on wood). No idea what did the trick, here's a list of the things I did in case it helps someone else:
*Installed 1 extra fan in my desktop, and replaced another one with a bigger version (GPU temperature is lower now)
*Updated everything: Windows, AMD drivers, motherboard drivers, sound card drivers, .NET Framework etc., the works.
*Set 3D Application Settings of AMD Catalyst Control Center to 'Performance'
*Disabled Origin In Game
*Running Inquisition in a borderless window
*Set Graphics in Inquisition to Medium, and changed Post-Process Quality to low
