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We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that typical turbulent accelerations of fluid particles are much smaller than gravity. This was shown to imply that the particles interact weakly with individual vortices and, as a result, form a smooth flow in most of the space. In weakly intermittent turbulence with moderate Reynolds number, rare regions where the flow breaks down can be neglected in the calculation of space averaged rate of droplet collisions. However, increase of Re increases probability of rare, large quiescent vortices whose long coherent interaction with the particles destroys the flow. Thus at higher Re the space averaged collision rate forms in rare regions where the assumption of smooth flow breaks down. We describe the transition between the regimes and provide collision kernel in the case of moderate Re describable by the flow. The distribution of pairwise distances is shown to obey a separable dependence on the magnitude and the polar angle of the separation vector. Magnitude dependence obeys a power-law with a negative exponent, manifesting multifractality of the droplet's attractor. We provide the so far missing numerical confirmation of a relation between this exponent and the Lyapunov exponents and demonstrate that it holds beyond the theoretical range. The angular dependence of the RDF exhibits a maximum at small angles quantifying particle's formation of spatial columns. We derive the droplet's collision kernel using that in the considered limit the gradients of droplet's flow are Gaussian. We demonstrate that as Re increases the column's aspect ratio decreases, eventually becoming one when the isotropy is restored. We propose how the theory could be constructed at higher Re of clouds by using the example of the RDF.