Understanding the intricacies of factors influencing European Union Emission Trading System (EU ETS) market prices is paramount for effective policy making and strategy implementation. We propose the use of the Information Imbalance, a non-parametric measure recently introduced in the physics community for quantifying the degree to which a set of variables is informative with respect to another one, to study the relationships among macroeconomic, economic, uncertainty, and energy variables concerning EU ETS price between January 2014 and April 2023. Our analysis shows that in Phase 3, commodity-related variables such as the ERIX index are the most informative in explaining the behaviour of the EU ETS market price. Transitioning to Phase 4, financial fluctuations take centre stage, with the uncertainty in the EUR/CHF exchange rate emerging as a crucial determinant. These results reflect the disruptive impacts of the COVID-19 pandemic and the energy crisis in reshaping the importance of the different variables. In addition to highlighting the shift in influential factors between Phase 3 and Phase 4, our findings underscore how macroeconomic volatility and energy disruptions have altered market dynamics. Notably, during the COVID-19 pandemic, the volatility in financial markets and fluctuations in energy demand and supply significantly affected the predictive power of different variables. Moreover, the energy crisis amplified the sensitivity of EU ETS prices to energy-related factors, reinforcing the importance of incorporating multiple dimensions into market analysis. Beyond variable analysis, we also propose to leverage the Information Imbalance to address the problem of mixed-frequency forecasting, and we identify the weekly time scale as the most informative for predicting the EU ETS price. Finally, we show how the Information Imbalance can be effectively combined with Gaussian Process regression for efficient nowcasting and forecasting using very small sets of highly informative predictors.