The lack of HDMI port support specification is sad from notebook manufacturers. From what you have done so far, your tv and external monitor cannot see the "expected" signal coming out from the notebook port, thus cannot communicate back to it correctly.
In your case I would try with different HDMI cables 1.3c or 1.4 per example, and contact ASUS for the specifics regarding devices supporting the current HDMI port. Though they may argue that the HDMI port is meant for video
playback only (hair splitting)