A quadrature element is an element used to represent a quantity at quadrature points. It does not have a set of underlying basis functions.
The family “Q” is an alternative to writing “CG” or “Lagrange”, and are defined as: DefElement: Lagrange
I dont understand this question. The FEniCS book was created back in 2012. The FEniCS tutorial was released in 2016 (compatible with version 2016.1.0 of dolfin). The FEniCSx tutorial was released in 2020, and is up to date with version 0.6.0 of DOLFINx. It has been recommended since Q2 of 2022 that people using fenics for new projects should use DOLFINx, see The new DOLFINx solver is now recommended over DOLFIN