We introduce a class of physically intriguing PT-symmetric Dirac-δ-Scarf-II optical potentials. We find the parameter region making the corresponding non-Hermitian Hamiltonian admit the fully real spectra, and present the stable parameter domains for these obtained peakons, smooth solitons, and double-hump solitons in the self-focusing nonlinear Kerr media with PT-symmetric δ-Scarf-II potentials. In particular, the stable wave propagations are exhibited for the peakon solutions and double-hump solitons from some given parameters even if the corresponding parameters belong to the linear PT-phase broken region. Moreover, we also find the stable wave propagations of exact and numerical peakons and double-hump solitons in the interplay between the power-law nonlinearity and PT-symmetric potentials. Finally, we examine the interactions of the nonlinear modes with exotic waves, and the stable adiabatic excitations of peakons and double-hump solitons in the PT-symmetric Kerr nonlinear media. These results provide the theoretical basis for the design of related physical experiments and applications in PT-symmetric nonlinear optics, Bose-Einstein condensates, and other relevant physical fields.