Your questions make me feel like a contestant in the Miss America Pageant. Would you accept "world peace" as my answer? Probably not. I must say you’re asking some very good — and very specific — questions, using terminologies that are becoming more widespread as LED use has increased, but are not yet in the common vernacular.
Let's start with CCT. Here is how the Illuminating Engineering Society (IESNA) describes it: The correlated color temperature (CCT) is a specification of the color appearance of the light emitted by a lamp, relating its color to the color of light from a reference source when heated to a particular temperature, measured in degrees Kelvin (K). The CCT designation for a light source gives a good indication of the lamp's general appearance, but does not give information on its specific spectral power distribution. Therefore, two lamps may appear to be the same color, but their effects on object colors can be quite different.
The spectral make-up of a light source affects its ability to render colors "naturally". Incandescent light has longer wavelengths (above 650 nm) of the visible spectrum and will render red colors most effectively. Certain common LEDs and fluorescents have more power in the short wavelength of the visible spectrum (below 450 nm), so blues will appear more vivid. Consequently, it comes down to what you are lighting and whether you want to enhance warm tones or cool tones.
Now to answer your question about the minimum value of CRI that would be acceptable to me in my lighting design. In theory, I should want the highest CRI possible, but since the lamp with the highest CRI is incandescent, it means that the color rendering qualities are skewed towards the warmer end of the spectrum.
In lighting kitchens, these days most of these rooms are part of an open plan concept where the kitchen visually flows into the family room, dining room or great room. Therefore, I would choose a high-efficacy light source with the higher CRI in order for these areas to appear to be unified. In California, Title 24 dictates that 50 percent of the lighting in the kitchen must come from high-efficacy sources.
However, I like to have a color temperature closer to daylight when lighting closets and laundry rooms, where color matching is important. These sources will have a lower CRI, and I'm okay with that. If I'm using high-efficacy sources in an environment that also includes incandescent light, then I would like to see a CRI of 90 or higher. When lighting closets, laundry rooms and landscape lighting in particular, I'm comfortable with a CRI of 80 to 85.
More and more, I am using hard-wired LED recessed adjustable fixtures in the other rooms of the house, so color matching is not an issue since the light sources are the same. That being said, I tend to add a warming filter that brings the color temperature from 2700K down to 2400K to equate the color of dimmed incandescent.
I feel that the general public will be more open to the idea of using alternative light sources if there is an option for the color of the light to be closer to 2400K, even if it means a lower lumen output. Since LEDs do not get warmer in color when dimmed, my solution is to create a color that is close to that of dimmed incandescent. Now we are seeing lamp and fixture manufacturers offering this lower color temperature as a standard option. Other companies, like Juno Lighting , Environmental Lights  and Philips Consumer Luminaires  are creating fixtures with both warm and cool colored LEDs that turn off in sequence when dimmed to create the illusion of going from a cooler color temperature to a warmer color temperature.