  We derive $ab$ $initio$ low-energy effective Hamiltonians (LEH) for
high-temperature superconducting (SC) copper oxides Bi$_2$Sr$_2$CuO$_6$
(Bi2201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 10$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$
(Bi2212, $N_{\ell}=2$, $T_c^{\rm exp} \sim 84$ K), HgBa$_2$CuO$_4$ (Hg1201,
$N_{\ell}=1$, $T_c^{\rm exp} \sim 90$ K) and CaCuO$_2$ (Ca11,
$N_{\ell}=\infty$, $T_c^{\rm exp} \sim 110$ K), with different experimental
optimal SC transition temperature $T_c^{\rm exp}$ and number $N_{\ell}$ of
laminated CuO$_2$ planes between the two neighboring block layers. We apply the
latest methodology of the multiscale $ab$ $initio$ scheme for correlated
electron systems (MACE), and focus on the LEH consisting of one antibonding
(AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$ orbital centered on each Cu atom. We
discuss prominent features of this LEH: (1) The ratio $U/|t_1|$ between the
onsite effective Coulomb repulsion (ECR) $U$ and amplitude of nearest neighbour
hopping $t_1$ increases with $T^{\rm exp}_c$ and $N_{\ell}$, consistently with
the expected increase in $d$-wave SC correlation function $P_{dd}$ with
$U/|t_1|$. One possible cause of the increase of $U/|t_1|$ is the replacement
of apical O atoms by Cu atoms from neighbouring CuO$_2$ planes when $N_{\ell}$
increases. Furthermore, we show that the increase in distance between Cu and
apical O atoms decreases the effective screening (ES) by electrons outside of
the LEH and increases $U/|t_1|$. (2) For Hg1201 and Ca11, we show that
$U/|t_1|$ decreases when hole doping per AB orbital $\delta$ increases, which
may partly account for the disappearance of SC when $\delta$ exceeds the
optimal value in experiment. (3) For $N_{\ell} \geq 2$, off-site inter-CuO$_2$
plane ECR is comparable to off-site intra-CuO$_2$ plane ECR. We discuss
contributions of inter-CuO$_2$ plane ECR to both $P_{dd}$ and the stability of
the SC state.
