MacBook Pro (Retina, Mid 2012) Freezes/Lags when connected to TV via HDMI
One thing you haven't tried is resetting the System Management Controller (SMC). This is something I would definitely do, especially because of the fan speeds and high heat. However, even though you've already reset the NVRAM, I would do that first again and then reset the SMC. Below are the steps I recommend for doing both resets on your model of MBP.

Reset the NVRAM

Older Macs had what's called Parameter RAM (PRAM), newer Macs (such as yours)  use Non-Volatile Random-Access Memory (NVRAM). I would reset this again before resetting the SMC. Here’s how to:


Shut down your machine. Yes, a full shut down, not just logging out.
Ensure you have nothing connected to your MBP (TV, external Keyboard, etc) - however you can keep the power cable plugged in.
Press the power button and then press the commandoptionpr keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
Hold those keys down until your Mac reboots again and you here the startup chime.
Let go of the keys and let your Mac reboot normally.


Note: When you log back in you may need to readjust some of your system preferences (e.g. speaker volume, screen resolution, startup disk selection, time zone information, etc).

Reset the SMC


Shut down your Mac
Keep the MagSafe adapter (power cable) plugged in.
Press at the same time shiftoptioncontrol (on the left side of the built-in keyboard) and the power button
Let go
Turn your computer back on with the power button.


After resetting both the NVRAM and SMC test to see if your MBP functions normally/better and let me know how you go.

Other factors to consider

Officially, your MBP is capable of an external resolution up to 2560 x 1600 pixels. So you will need to ensure that you're using the right settings to accommodate that.