Yes, a neutral tint has better color rendition. This is a fact. There is much documentation about color spectrum and wave lengths of neutral tint vs cool tint. This does not mean everyone likes netral better. Many still like cool tinted lights better. It seems people with more lights eventually prefer neutral.
This isn't true. (However, in the case of LEDs, it's almost always true. Perhaps this is what you meant?) If you consider daylight [sunlight], which is ~5500-6500K "neutral/cool" white - it gives very good color rendition because of its smooth and continuous spectrum.
You could have a neutral light source made up of just a few spikes in a spectrum - which would give you the tint, but not the color rendition. Basically, when it comes to color rendition, you should take into account both tint, as well as the spectrum (which tells you a lot more than a simple "xx CRI" measurement).
(Example: In the case of Cree, they have both neutral LEDs and outdoor LEDs - both have neutral tint bins, but the neutral ones have better color rendition due to the different phosphors used)
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As for "neutral", I dislike using the term. Everyone has different ideas on what it means, so I prefer to just say actual CCTs like "5000K" or "6000K". I'd define neutral as whatever color temperature an individual regards as "pure white", which is different for everyone.