Oil and gas production in permafrost can effectively alleviate energy tensions. However, ice melting around wellbores and the accompanying wellhead instability affect the efficiency and safety of oil and gas development in permafrost. Moreover, the potential oil and gas leakage will damage the environment and the ecology of permafrost. Unfortunately, ice melting, formation subsidence, and wellhead behavior during this process have rarely been investigated in previous studies. In the present work, mechanical properties of permafrost were first experimentally investigated, which provided the basic parameter for subsequent numerical simulation. It was found that the ultimate strength gradually increased with the decreasing temperature, as well as the increasing confining pressure. Meanwhile, although the elastic modulus increased with decreasing temperature, it was less affected by confining pressure. Unlike other parameters, the Poisson’s ratio was hardly affected by temperature and confining pressure. Moreover, both the internal friction angle and the cohesion increased with decreasing temperature, but the influence degree varied within different temperature ranges. Then, ice melting, formation subsidence, and the instability behavior of the wellhead caused by the disturbance of the development operation were numerically explored. The investigation results show that the ice melting range in the reservoir section reached 8.06 m, which is much wider than that in other well sections. Moreover, failure of the cement–permafrost interface, caused by ice melting, resulted in a wellhead sinking of up to 1.350 m. Finally, the insulation effect of the vacuum-insulated casing showed that the temperature drop of the designed vacuum-insulated casing was much lower than that of the ordinary casing. When the fluid temperature within the wellbore was 70 °C, the temperature drop of the designed vacuum-insulated casing was 3.54 °C lower than that of the ordinary casing. This study provides support for maintaining wellhead stability during oil and gas extraction in permafrost for avoiding some environmental disasters (such as oil and gas leakage).