This paper investigates a class of McKean-Vlasov stochastic neural networks (MV-SNNs) subject to Lévy noise, mixed time-varying delays, and coefficients that depend on both the state and its probability distribution. Existing methods struggle to concurrently address the solution’s existence and uniqueness, long-term dynamics analysis, and the construction of stability criteria. To tackle this, we develop a systematic analytical framework via an innovative methodology that combines fixed point theory with the resolvent operator (RO) technique. First, novel criteria for the existence and uniqueness of mild solutions are established. Second, by integrating the RO with Hölder’s inequality and Itô’s isometry, sufficient conditions for the global attracting set (GAS) and quasi-invariant set (QIS) are derived, enabling quantitative analysis of long-term dynamics. Furthermore, using the RO together with Gronwall’s inequality, Chebyshev’s inequality, and the Borel-Cantelli lemma, we establish criteria for mean square and almost sure exponential stability. These moment stability criteria inherently guarantee solution existence and uniqueness without requiring additional constraints. Finally, two examples demonstrate the applicability and effectiveness of the theoretical results.