As the limit equations of mean-field particle systems perturbed by common environmental noise, the McKean-Vlasov stochastic differential equations with common noise have received a lot of attention. Moreover, past dependence is an unavoidable natural phenomenon for dynamic systems in life sciences, economics, finance, automatic control, and other fields. Combining the two aspects above, this paper delves into a class of nonlinear McKean-Vlasov stochastic functional differential equations (MV-SFDEs) with common noise. The well-posedness of the nonlinear MV-SFDEs with common noise is first demonstrated through the application of the Banach fixed-point theorem. Secondly, the relationship between the MV-SFDEs with common noise and the corresponding functional particle systems is investigated. More precisely, the conditional propagation of chaos with an explicit convergence rate and the stability equivalence are studied. Furthermore, the exponential stability, an important long-time behavior of the nonlinear MV-SFDEs with common noise, is derived. To this end, the Itô formula involved with state and measure is developed for the MV-SFDEs with common noise. Using this formula, the Razumikhin theorem is proved, providing an easy-to-implement criterion for the exponential stability. Lastly, an example is provided to illustrate the result of the stability.