The properties of toroidal hyperheavy even-even nuclei and the role of toroidal shell structure are extensively studied within covariant density functional theory. The general trends in the evolution of toroidal shapes in the $Zapprox 130-180$ region of nuclear chart are established for the first time. These nuclei are stable with respect of breathing deformations. The most compact fat toroidal nuclei are located in the $Zapprox 136, Napprox 206$ region of nuclear chart, but thin toroidal nuclei become dominant with increasing proton number and on moving towards proton and neutron drip lines. The role of toroidal shell structure, its regularity, supershell structure, shell gaps as well as the role of different groups of the pairs of the orbitals in its formation are investigated in detail. The lowest in energy solutions at axial symmetry are characterized either by large shell gaps or low density of the single-particle states in the vicinity of the Fermi level in at least one of the subsystems (proton or neutron). Related quantum shell effects are expected to act against the instabilities in breathing and sausage deformations for these subsystems. The investigation with large set of covariant energy density functionals reveals that substantial proton $Z=154$ and 186 and neutron $N=228$, 308 and 406 spherical shell gaps exist in all functionals. The nuclei in the vicinity of the combination of these particle numbers form the islands of stability of spherical hyperheavy nuclei. The study suggests that the $N=210$ toroidal shell gap plays a substantial role in the stabilization of fat toroidal nuclei.
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