  Competition and cooperation between electron correlation and relativistic
spin-orbit coupling give rise to diverse exotic quantum phenomena in solids. An
illustrative example is spin-orbit entangled quantum liquids, which exhibit
remarkable features such as topological orders and fractional excitations. The
Kitaev honeycomb model realizes such interesting states, called the Kitaev spin
liquids, but its experimental feasibility is still challenging. Here we
theoretically investigate hexagonal heterostructures including a candidate for
the Kitaev magnets, MgIrO$_3$, to actively manipulate the electronic and
magnetic properties toward realizing the Kitaev spin liquids. For three
different structure types of ilmenite bilayers MgIrO$_3$/$A$TiO$_3$ with $A$ =
Mn, Fe, Co, and Ni, we obtain the optimized lattice structures, the electronic
band structures, the stable magnetic orders, and the effective magnetic
couplings. We find that the spin-orbital coupled bands characterized by the
pseudospin $j_{\rm eff}=$ 1/2 are retained in the MgIrO$_3$ layer for all the
heterostructures, but the magnetic state and the band gap depend on the types
of heterostructures as well as the $A$ atoms. In particular, one type becomes
metallic irrespective of $A$, while the other two are mostly insulating. We
show that the insulating cases provide spin-orbit coupled Mott insulating
states with dominant Kitaev-type interactions, accompanied by different
combinations of subdominant interactions depending on the heterostructural type
and $A$, while the metallic cases realize spin-orbit coupled metals with
various doping rates. Our results indicate that these hexagonal
heterostructures are a good platform for engineering electronic and magnetic
properties of the spin-orbital coupled correlated materials, including the
possibility of Majorana Fermi surfaces and topological superconductivity.
