The “Intel HD Audio” option appears in the Sound Properties only when an HDMI or DisplayPort device with audio capabilities is properly detected.
In your case no device is being detected in that area and this could be because the DisplayPort is faulty, the DP-to-HDMI adapter is not working properly, or the adapter is not compatible.
I am wondering what is listed under “Monitors” in Device Manager. Is it a generic PnP Monitor or it shows the actual brand and model of your display? If it shows a generic PnP monitor then your monitor is not detected properly.
You already covered the possible options to check. Making sure the DP port is enabled in the BIOS and updating the Intel® HD Graphics driver are the most important steps. You can still update the motherboard BIOS just in case that helps.
Before updating the motherboard BIOS I would suggest first to confirm that the DisplayPort works fine. If it is possible, test a straight DP device (TV or AVR). If the audio and video come up fine with a straight DP connection then the problem is the adapter.
Bear in mind that the Intel® Desktop Board DQ67SW is a product released in 2011 and it will be easy to find compatibility problems with devices of more recent manufacturing.
To update the BIOS, follow the following instructions: http://www.intel.com/support/motherboards/desktop/sb/CS-030144.htm
Download the latest BIOS file (v0067) using this other link: