I found the manual for your TV. I now have a much better idea of what we're dealing with, and I know how to fix HDMI once and for all (I hope).
HDMI on computers is a mess, and always has been.
With VGA, the HDTV acts like a computer monitor, and the computer thinks the HDTV is a computer monitor. This makes everything work ok, but at the cost of image quality.
With HDMI, the HDTV thinks the computer is a cable box, and the computer thinks the HDTV...is an HDTV. This makes everything wrong by default.
When both are set up correctly, you will get the best possible quality from that TV.
If you ever feel like giving up and sticking with DVI-to-VGA, that's fine. Whatever works for your setup will honestly be good enough.
If you want to pull through and set up HDMI properly, these are the objectives:
1. Fix the screen resolutions. HDTVs work best when handling their native resolution, which is 1080p in this case. Therefore, force your GPU to always output 1080p to the TV, even when playing lower-resolution games.
2. Fix the TV. The TV is acting like a TV and not a computer monitor. We need to change that. This should also fix the input lag.
You will need to install the latest Nvidia drivers; it's the only way to make sure the settings are correct.
Connect your computer to HDMI 2. This is the special "PC" port for HDMI on your TV.
Pull up a chair and set your desktop to full 1080p. You can change the resolution to 1366x768 later.
You might not be able to see the entire Windows desktop (start button, etc) due to cropping. This is a good thing, we'll fix that later.
Open the Nvidia Control Panel. Go to "Adjust desktop size and position"
Select a scaling mode: Aspect Ratio
Perform scaling on: GPU
Check "Override the scaling mode set by games and programs."
Resolution: 1920x1080
Refresh rate: 60Hz
Uncheck "Enable Desktop Resizing".
Apply settings. Use the INFO button, ensure the TV is displaying 1080p @ 60Hz.
Also make sure the PC is outputting Full Range colors.
Open your TV menu. Go down to the "Input" menu --> Edit Name --> HDMI2. Set the input name to "PC". This step is crucial; it should remove cropped screen edges, reduce input lag, and reconfigure the colors.
Back out of the TV menu and visit the Picture settings. Change as many of the following as possible:
Mode: Standard
Backlight: Whatever you want.
Contrast: 100, or as high as you can go
while still being able to see Color 254 on this page.
Brightness: 45
Sharpness: 0
Color: 50
Tint: 50
Open the "Detailed Settings" if you can. Change as many of the following as possible:
Black Adjust: Dunno. "High" I guess, just make sure
you can still see all the black squares on this page and that the background is as black as possible. You might have to readjust the brightness/contrast after this.
Dynamic Contrast: Off
Gamma: 0
Color Space: Native
White Balance: Reset
Flesh Tone: 0
Edge Enhancement: Off
xvYCC: Off
Please let me know which settings were disabled, if any.
At this point, you can change your desktop resolution back to 1366x768. Yet, when you press the INFO button on your remote; it should say 1920x1080.
Play a Touhou game, it should still say 1920x1080, and it will be in uncropped 4:3.
Everything should now work visually.
Btw unplug any 3.5mm audio cables from the computer and make sure the sound works over HDMI. If it doesn't, that's another silly quirk and an easy tix (in theory).
Edit: Some clarifications.