I'm going to walk you through some basic troubleshooting steps to eliminate any possible PEBCAK issues. Please take no offense as I believe they will resolve your problem. To begin, unplug the HDMI cable from your laptop. Verify it is inserted fully into the display you wish to use. If uncertain, hook up another device to make sure it works. Next, in your Windows 8.1 environment, please download and install the following drivers. They are directly from Asus, for your model.
Perform a clean install for the nVidia driver.
After you've installed and restarted the PC, insert the HDMI cable fully into the laptop. Some models feel snug before full contact is made. If you can still clearly see bare metal, it is probably not in all the way. At this time turn on the television and set it to the correct channel for the HDMI port you wish to use.
As you may have guessed, windows + P can be used to quickly extend or duplicate your desktop. Right clicking your desktop and navigating to advanced display properties will bring up all of your possible connection methods. Alternatively, select the NVIDIA Control Panel. If the display port is listed but the display itself is not, I recommend changing the HDMI cable. If a second display is not listed there, you may have an incorrect BIOS value orpossibly a hardware failure. However, based on your DXDiag, I suspect the culprit will be resolved by the above.
EDIT: As you have the same problem in Linux, you can probably skip the driver reinstall and move straight to the cable and channel diagnosis. Make sure that the HDMI cable AND that specific HDMI port on the TV work correctly, and that the TV is on with that channel selected.
You do have both Intel and nVidia graphics, this is a dual-chip design called "Optimus" which uses the Intel integrated graphics primarily for low power and heat operation, while having the nVidia chip to switch to on-the-fly for more power demanding operations, such as gaming. The process is automatic and requires no user interaction, so long as the drivers are properly configured.