I saw a talk recently mentioning that another factor that may determine a star's metallicity (besides age) may be how far from the center of its "birth cluster" it is when the short lived hot central stars of the cluster go supernova. I wonder if someone could comment on this. I also wonder if, since now we have better data, have we found more of a continuum of metallicities or is the "Population I, Population II, Population III" idea still going strong??
The idea of self-pollution in a cluster may be viable. When you talk about metallicity, there are two things - one is what you see, the photospheric metallicity; the second is the bulk metallicity.
To alter the former only requires that adjacent stars accrete enriched material and then that they are unable to mix and dilute this. The second is much harder. If you want to form a star with high metallicity, the material it forms from must be enriched first, e.g. a supernova progenitor must have lived and died first. As the earliest age that a supernova can explode at is about 5 million years, this means having an extended period of star formation in a single cluster. This is controversial. There is a lot of evidence that young star clusters form very quickly, in which case the supernova mechanism is unviable. Nearby open clusters also show no evidence of star-to-star composition variations.
However, ancient globular clusters may be different. There is evidence for multiple populations with differing compositions.
Your final question is too broad. Yes, there are many nuances and gradations in populations, though the catch-all terms of Populations I and II are still used. But for example we also talk about thin and thick disc populations or bulge populations.