Essentially, when you switch to ATA, you are switching to a different SATA controller than the one your are currently using, and Windows does not have a driver for it, hence the blue screen when you try to boot. By installing a SATA card, you are providing a bootstrap device that will allow Windows to boot, after which you can install the drivers for the onboard SATA. Once those drivers are installed, you no longer need a bootstrap device and you can reconnect the boot drive to the onboard port and Windows will boot properly once again.
I used a 960 last year, but I don't remember it needing any special attention for two storage devices. (I had it set up with an SSD for boot and a hard drive for storage.)
I'm wondering if Intel RST has something funky configured that is causing the drives to be seen as a RAID set.