The EU carbon market serves as an innovative financial instrument with the primary objective of contributing to mitigate the impacts of climate change. This market demonstrates significant interconnectedness with fossil energy, precious metal, and financial markets, although limited research has focused on the causality, dependency, intensity and direction of time-varying spillover effects. This study aims to investigate the causality direction, degree of dependency structure, and volatility transmission from Brent Oil, UK Natural Gas, Rotterdam Coal, Gold, Silver, Copper, and EuroStoxx600 future prices to EU Allowances during different periods of EU market. To achieve these objectives, this paper proposes a novel methodological approach that combines the most recent econometrics methods, such as Directed Acyclic Graph analysis, C-Vine Copula models, and Time-Varying parameter Vector Auto Regressive models with Stochastic Volatility with the use of a comprehensive sample of daily data from 26 April 2005 to 31 December 2022. The major findings of this study demonstrate that causality predominantly runs from energy, metal, and financial markets to the EU carbon market. The dependency structure, although varying across different sub-periods, shows a strong relationship observed between oil, coal, silver, copper, EuroStoxx600, and CO2 market. Additionally, the oil and copper futures prices exhibit the highest dependence on EUA prices. Furthermore, the study establishes that the EU carbon market is a net receiver of shocks from all other markets, with the energy, metal, and financial markets significantly influencing volatility in EUA prices. The time-varying spillover effect is most pronounced with a one-day lag, and the duration of the spillover effects ranges from 2 to 15 days, gradually diminishing over time. These results have the potential to increase the understanding of the EU carbon market and offer practical guidance for policymakers, investors, and companies involved in this domain.