The Dragonfly Galaxy (MRC 0152-209), the most infrared-luminous radio galaxy at redshift z~2, is a merger system containing a powerful radio source and large displacements of gas. We present kpc-resolution data from ALMA and the VLA of carbon monoxide (6-5), dust, and synchrotron continuum, combined with Keck integral-field spectroscopy. We find that the Dragonfly consists of two galaxies with rotating disks that are in the early phase of merging. The radio jet originates from the northern galaxy and brightens when it hits the disk of the southern galaxy. The Dragonfly Galaxy therefore likely appears as a powerful radio galaxy because its flux is boosted into the regime of high-z radio galaxies by the jet-disk interaction. We also find a molecular outflow of (1100 $\pm$ 550) M$_{\odot}$/yr associated with the radio host galaxy, but not with the radio hot-spot or southern galaxy, which is the galaxy that hosts the bulk of the star formation. Gravitational effects of the merger drive a slower and longer lived mass displacement at a rate of (170 $\pm$ 40) M$_{\odot}$/yr, but this tidal debris contain at least as much molecular gas mass as the much faster outflow, namely M(H2) = (3 $\pm$ 1) x 10$^9$ (alpha(CO)/0.8) M$_{\odot}$. This suggests that both the AGN-driven outflow and mass transfer due to tidal effects are important in the evolution of the Dragonfly system. The Keck data show Ly$\alpha$ emission spread across 100 kpc, and CIV and HeII emission across 35 kpc, confirming the presence of a metal-rich and extended circumgalactic medium previously detected in CO(1-0).