Unconventional superconductivity in molecular conductors is observed at the border of metal-insulator transitions in correlated electrons under the influence of geometrical frustration. The symmetry as well as the mechanism of the superconductivity (SC) is highly controversial. To address this issue, we theoretically explore the electronic properties of carrier-doped molecular Mott system $kappa$-(BEDT-TTF)$_2$X. We find significant electron-hole doping asymmetry in the phase diagram where antiferromagnetic (AF) spin order, different patterns of charge order, and SC compete with each other. Hole-doping stabilizes AF phase and promotes SC with $d_{xy}$-wave symmetry, which has similarities with high-$T_c$ cuprates. In contrast, in the electron-doped side, geometrical frustration destabilizes the AF phase and the enhanced charge correlation induces another SC with extended-$s$+$d_{x^2-y^2}$-wave symmetry. Our results disclose the mechanism of each phase appearing in filling-control molecular Mott systems, and elucidate how physics of different strongly-correlated electrons are connected, namely, molecular conductors and high-$T_c$ cuprates.
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