The Linac Coherent Light Source II (LCLS-II) is designed to generate 4 GeV electron bunches at an up-to-1-MHz high repetition rate to drive X-ray Free Electron Lasers. It has been in user operation since 2023 and has progressively ramped up the beam repetition rate to 93 kHz to date. The LCLS-II 1-MeV electron source was successfully commissioned between 2018 and 2020. The full-scale LCLS-II injector, operating above 75 MeV, began commissioning in 2022 and transitioned to operations a few months later. The LCLS-II injector has demonstrated the ability to deliver a high-brightness, low-emittance electron beam at high repetition rates, meeting key performance targets. However, several operational challenges have emerged during commissioning and early user operations. These include substantial gun dark current, the dependence of bunch charge generation and free-electron laser pulse intensity on the beam repetition rate, the formation of cathode quantum efficiency craters and bumps, significant emittance growth through the laser heater chicane, beam breakup, and the presence of beam halos. This paper presents recent progresses in addressing these challenges, along with injector emittance measurements and optimizations, and the first demonstration of operating laser heater undulator at the third-harmonic resonance.