There's a few things to consider. - Resolution/pixel density. The more dots you have on a certain area the closer you can watch without seeing pixels instead of details. Then again, the higher the resolution the better video card you'd need.
- Pixel size. If you like the looks of Minecraft, large pixels aren't an issue. Your own eyes are the best tool here.
- Input lag, which means how long it takes for the TV to show an image processed in the video card. Some TV's work faster with computers than others, there's charts in the Internet about it. You don't want to be half a dozen frames behind what's really happening!
- Compatibility. Although you can connect basically any TV to any computer, you may not necessarily get the refresh rate you'd need for maximum FPS. Especially with 4K you'd need both a strong video card and a certain type/generation of cable to get past 30 Hz/FPS.
If your 65" TV and your computer have a matching pair of connectors, you can't break anything by just hooking them together. If the result pleases you, the most important player will be happy. 
Excellent post..Bizman , especially #4
Edit: RE reading your list, I would actually invert it 4 = 1, 3 = 2, 2 = 3, 1 = 4...
~S~
TC