The first stars: CEMP--no stars and signatures of spinstars

(Abridged) The CEMP--no stars are "carbon-enhanced-metal-poor" stars that in principle show no evidence of s-- and r--elements from neutron captures. We try to understand the origin and nucleosynthetic site of their peculiar CNO, Ne--Na, and Mg--Al abundances. We compare the observed abundances to the nucleosynthetic predictions of AGB models and of models of rotating massive stars with internal mixing and mass loss. We also analyze the different behaviors of $\alpha$-- and CNO--elements, as well the abundances of elements involved in the Ne--Na and Mg--Al cycles. We show that CEMP-no stars exhibit products of He--burning that have gone through partial mixing and processing by the CNO cycle, producing low $^{12}$C/$^{13}$C and a broad variety of [C/N] and [O/N] ratios. From a $^{12}$C/$^{13}$C vs. [C/N] diagram, we conclude that neither the yields of AGB stars (in binaries or not) nor the yields of classic supernovae can fully account for the observed CNO abundances in CEMP-no stars. Better agreement is obtained once the chemical contribution by stellar winds of fast-rotating massive stars is taken into account, where partial mixing takes place, leading to various amounts of CNO being ejected. CEMP--no stars present a wide variety in the [C/Fe], [N/Fe], [O/Fe], [Na/Fe], [Mg/Fe], [Al/Fe], and [Sr/Fe] ratios. We show that back-and-forth, partial mixing between the He-- and H--regions may account for this variety. Some s--elements, mainly of the first peak, may even be produced by these processes in a small fraction of the CEMP--no stars. We propose a classification scheme for the CEMP--no and low--s stars, based on the changes in composition produced by these successive back-and-forth mixing motions.