We performed systematic studies of the combined effects of annealing/quenching temperature ({itshape T}$_{A/Q}$) and T = Ni, Rh substitution ({itshape x}) on the physical properties of Ca(Fe$_{1-x}$T$_{x}$)$_{2}$As$_{2}$. We constructed two-dimensional, {itshape T}$_{A/Q}$-{itshape x} phase diagrams for the low-temperature states for both substitutions to map out the relations between ground states and compared them with that of Co-substitution. Ni-substitution, which brings one more extra electron per substituted atom and suppresses the {itshape c}-lattice parameter at roughly the same rate as Co-substitution, leads to a similar parameter range of antiferromagnetic/orthorhombic in the {itshape T}$_{A/Q}$-{itshape x} space as that found for Co-substitution, but has the parameter range for superconductivity shrunk (roughly by a factor of two). This result is similar to what is found when Co- and Ni-substituted BaFe$_{2}$As$_{2}$ are compared. On the other hand, Rh-substitution, which brings the same amount of extra electrons as does Co-substitution, but suppresses the {itshape c}-lattice parameter more rapidly, has a different phase diagram. The collapsed tetragonal phase exists much more pervasively, to the exclusion of the normal, paramagnetic, tetragonal phase. The range of antiferromagnetic/orthorhombic phase space is noticeably reduced, and the superconducting region is substantially suppressed, essentially truncated by the collapsed tetragonal phase. In addition, we found that whereas for Co-substitution there was no difference between phase diagrams for samples annealed for one or seven days, for Ni- and Rh- substitutions a second, reversible, effect of annealing was revealed by seven-day anneals.
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