Novel superhard tetragonal carbon allotropes C$_5$, C$_6$, and C$_7$, characterized by the presence of sp$^3$- and sp$^2$-like carbon sites, have been predicted from crystal chemistry and extensively studied by quantum density functional theory (DFT) calculations. All new allotropes were found to be cohesive, with crystal densities and cohesive energies decreasing along the C$_5$-C$_6$-C$_7$ series due to the greater openness of the structures resulting from the presence of C=C ethene and C=C=C propadiene subunits, and they were mechanically stable, with positive sets of elastic constants. The Vickers hardness evaluated by different models qualifies all allotropes as superhard, with Hv values ranging from 90 GPa for C$_5$ to 79 GPa for C$_7$. Phonon band structures confirm that the new allotropes are also dynamically stable. The electronic band structures reveal their metallic-like behavior due to the presence of sp$^2$-hybridized carbon.