We use Atacama Large Millimetre Array and Jansky Very Large Array observations of the galaxy cluster Cl J1449+0856 at z = 1.99, in order to study how dust-obscured star formation, interstellar medium content, and active galactic nuclei activity are linked to environment and galaxy interactions during the crucial phase of high-z cluster assembly. We present detections of multiple transitions of 12CO, as well as dust continuum emission detections from 11 galaxies in the core of Cl J1449+0856. We measure the gas excitation properties, star formation rates, gas consumption timescales, and gas-to-stellar mass ratios for the galaxies. We find evidence for a large fraction of galaxies with highly excited molecular gas, contributing >50 per cent to the total star formation rate (SFR) in the cluster core. We compare these results with expectations for field galaxies, and conclude that environmental influences have strongly enhanced the fraction of excited galaxies in this cluster. We find a dearth of molecular gas in the galaxies' gas reservoirs, implying a high star formation efficiency (SFE) in the cluster core, and find short gas depletion timescales τdep < 0.1-0.4 Gyr for all galaxies. Interestingly, we do not see evidence for increased specific SFRs (sSFRs) in the cluster galaxies, despite their high SFEs and gas excitations. We find evidence for a large number of mergers in the cluster core, contributing a large fraction of the core's total star formation compared with expectations in the field. We conclude that the environmental impact on the galaxy excitations is linked to the high rate of galaxy mergers, interactions, and active galactic nuclei in the cluster core.