In this paper, at the first time, we perform the analysis of correlational and non-extensive properties of the CO2 emission market relying on the carbon emissions futures time series for the period 04.07.2008-10.05.2021 and the daily data of the power sector from the U.S. Carbon Monitor for the period 01.01.2019-10.05.2021, which consist the data of both individual countries (USA, Germany, China, India, United Kingdom, et al.) and global emissions (World). For comparison, we present the analysis of the Dow Jones Industrial Average (DJIA) index. Our results show that both futures and the DJIA are presented to be non-extensive, and the distribution of their normalized returns can be better described by power-law probability distributions, particularly, by q-Gaussian. We estimate Tsallis triplet for the entire time series of CO2 emissions futures and the DJIA and present q-triplet as an indicator of crisis phenomena, relying on the sliding window algorithm. It can be seen that the triplet behaves in a characteristic way during economic crises. The toolkit of random matrix theory (RMT) allowed us to investigate the correlational nature of the carbon emissions market and to build appropriate indicators of crisis phenomena, which clearly reflect the collective dynamics of the entire research base during events of this kind.