Morphology-density Relation, Quenching, and Mergers in CARLA Clusters and Proto-Clusters at $1.4<z<2.8$

(Abridged) To understand if the morphology-density and passive-density relations are already established at z>1.5, we study galaxies in 16 confirmed clusters at $1.3<z<2.8$ from the CARLA survey. Our main finding is that the morphology-density and passive-density relations are already in place at $z\sim2$. The cluster at z = 2.8 shows a similar fraction of ETG as in the other clusters in its densest region. The cluster ETG and passive fractions depend on local environment and mildly on galaxy mass. They do not depend on global environment. At lower local densities, the CARLA clusters exhibit a lower ETG fraction than clusters at z = 1. This implies that the densest regions influence the morphology of galaxies first, with lower density local environments either taking longer or only influencing galaxy morphology at later cosmological times. Interestingly, we find evidence of high merger fractions in our clusters with respect to the field, but the merger fractions do not significantly depend on local environment. This suggests that merger remnants in the lowest density regions can reform disks fuelled by cold gas flows, but those in the highest density regions are cut-off from the gas supply and will become passive ETG. The percentages of active ETG, with respect to the total ETG population, are $21 \pm 6\%$ and $59 \pm 14\% $ at 1.35 < z <1.65 and 1.65 < z < 2.05, respectively, and about half of them are mergers or asymmetric in both redshift bins. All the spectroscopically confirmed CARLA clusters have properties consistent with clusters and proto-clusters. The differences between our results and those that find enhanced star formation and star-bursts in cluster cores at similar redshifts are probably due to the different sample selection criteria, which choose different environments that host galaxies with different accretion and pre-processing histories.