We study the decays B- --> D*0 pi- and B- --> D*0 K-, where the D*0 decays into D0 pi0, with the D0 reconstructed in the CP-even (CP+) eigenstates K- K+ and pi- pi+ and in the (non-CP) channels K- pi+, K- pi+ pi+ pi-, and K- pi+ pi0. Using a sample of about 123 million BBbar pairs, we measure the ratios of decay rates R*(non-CP)=BR(B- --> D*0(non-CP) K-)/BR(B- --> D*0(non-CP) pi-) = 0.0813+-0.0040(stat)+0.0042-0.0031}(syst), and provide the first measurements of R*(CP+)= BR(B- --> D*0(CP+) K-)/BR(B- --> D*0(CP+) pi-) = 0.086+-0.021(stat)+-0. 007(syst), and of the CP asymmetry A*(CP+) = (BR(B- --> D*0(CP+) K-) - BR(B+ --> D*0(CP+) K+))/(BR(B- --> D*0(CP+) K-) + BR(B+ --> D*0(CP+) K+)) = -0.10+-0.23(stat)+0.03-0.04(syst).