Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene (RTG) which observed spin-valley symmetry-breaking and superconductivity, we study instabilities of the RTG metallic state to symmetry breaking orders. We find that interactions select the inter-valley coherent order (IVC) as the preferred ordering channel over a wide range, whose theoretically determined phase boundaries agree well with experiments on both the hole and electron doped sides. The Fermi surfaces near van Hove singularities admit partial nesting between valleys, which promotes both inter-valley superconductivity and IVC fluctuations. We investigate the interplay between these fluctuations and the Hunds (intervalley spin) interaction using a renormalization group approach. For antiferromagnetic Hunds coupling, intervalley pairing appears in the spin-singlet channel with enhanced $T_c$, that scales with the dimensionless coupling $g$ as $T_csimexp(-1/sqrt{g})$ , compared to the standard $exp(-1/g)$ scaling. In its simplest form, this scenario assumes a sign change in the Hunds coupling on increasing hole doping. On the other hand, the calculation incorporates breaking of the independent spin rotations between valleys from the start, and strongly selects spin singlet over spin triplet pairing, and naturally occurs in proximity to the IVC, consistent with observations.
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