Self-starting self-pulsing dynamics at the cavity free-spectral-range frequency were observed in intracavity second-harmonic generation of a diode end-pumped Nd:YLF ring laser containing a periodically-poled KTiOPO(4) (ppKTP) nonlinear crystal. Although the unidirectional laser was designed for continuous-wave (cw) single-frequency operation, with a resonator set at the middle of its stability range, partial Kerr-lens mode-locking (KLM) arose spontaneously once the ppKTP was inserted. This ultrafast dynamics along with a strong spectral gain broadening, not observed with any birefringent nonlinear doubler, is associated to the finite bandwidth of the quasi-phase-matched crystal with respect to the laser gain bandwidth, leading to giant cascaded Kerr-lensing effects when the ppKTP temperature is detuned from perfect quasi-phase-matching either in the self-focusing or defocusing sides. While under partial KLM operation the laser delivered only approximately 0.14W of broadband red output power, single-frequency operation could be only achieved by using an intracavity etalon with a suitable partial reflectivity (R>or=25%), leading to an optimally (approximately100% efficiency) out-coupled 1.4 W red power at 660.5 nm, as much as the fundamental 1321 nm power that could be extracted from the unidirectional laser using an optimal T = 2% output coupler.