In the standard Overlapping Generations model of Diamond, there are two generations, each living two periods. In the first period of their life, the Young have no capital, and they work in firms where the capital is owned by the Old. So in the heart of, let's say, "western mainstream economics", we have a model that relies on the total separation of labor and capital, and distinguishes between "workers" and "capitalists" (in terms here of property rights) in the most clear-cut way.
This is a good example in my opinion as to why the term" "capitalist", if properly defined (as it must be done with all terms that have also sociopolitical flavor), can be used in economics of any strand and for proper economic modeling and thinking.
You are certainly right in pointing out that the question as posed by the OP was clumsy - and at the same time, as is sometimes the case with clumsy questions, it provided an opportunity to try to properly treat the term.
As for your three inquiries:
1) I don't remember seeing an economics model or book using the "nouveau riche" concept (as perhaps a label for special behavioral assumptions that could lead to special outcomes). Off-topic also because it does not involve economic inference.
2) This asks for opinion-based answers, and the re-phrase "is there any study as to how many people consider..." appears necessary. But it would be off-topic anyhow, since it does not involve economic inference.
3) In my opinion, this asks for opinion-based answers.