Does using a DP->HDMI adapter have 100% compatibility or are there known problems hooking up a TV?
The i5-4590T has integrated graphics with HDMI capability. How do I physically change the DP port to HDMI so that I don't need to use a DP->HDMI adapter?
You can't physically remove a DP port and replace it with a HDMI. You just need the right cable and it should work fine.