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I think that one of your questions remained unanswered: How does one create a polynomial ring with one variable that has the same class as a multivariate polynomial ring?

Indeed, Sage differentiates between a univariate ring (which has a specialised implementation) and a multivariate ring with one variable.

If you do

sage: R_univariate.<x> = QQ[]

then you will obtain a univariate ring, which has a different class than

sage: R_multivariate.<x,y> = QQ[]

You can obtain a multivariate ring with one variable as follows:

sage: R_one_variable.<x> = PolynomialRing(QQ,1)
sage: type(R_multivariate) == type(R_one_variable)
True