We report the growth of single crystals of SrCu2As2, SrCu2Sb2, SrCu2(As0.84Sb0.16)2 and BaCu2Sb2 using the self-flux technique and their structural, magnetic, thermal and transport properties that were investigated by powder x-ray diffraction (XRD), magnetic susceptibility χ, specific heat Cp and electrical resistivity ρ measurements versus temperature T from 1.8 to 350 K. Rietveld refinements of XRD patterns for crushed crystals confirm that SrCu2As2 crystallizes in the ThCr2Si2-type body-centered tetragonal structure (space group I4/mmm) and SrCu2Sb2 crystallizes in the CaBe2Ge2-type primitive tetragonal structure (space group P 4/nmm). However, as reported previously, BaCu2Sb2 is found to have a large unit cell consisting of three blocks. Here a ThCr2Si2-type block is sandwiched between two CaBe2Ge2-type blocks along the c-axis with an overall symmetry of I4/mmm, as reported, but likely with a monoclinic distortion. The χ data of all these compounds are diamagnetic and reveal nearly T -independent anisotropic behavior. The χ of SrCu2As2 is found to be larger in the ab-plane than along the c-axis, as also previously reported for pure and doped BaFe2As2, whereas the χ values of SrCu2Sb2 and BaCu2Sb2 are larger along the c-axis. This difference in anisotropy appears to arise from the differences between the crystal structures. The finite values of the Sommerfeld linear specific heat coefficients γ and the T dependences of ρ reveal metallic character of all four compounds. The electronic and magnetic properties indicate that these compounds are sp metals with Cu in the nonmagnetic 3d 10 electronic configuration corresponding to the oxidation state Cu +1 , as previously predicted theoretically for SrCu2As2 by D. J. Singh [Phys. Rev. B 79, 153102 (2009)]. We present a brief review of theoretical and experimental work on the doping character of transition metals for Fe in BaFe2As2. The As-As covalent interlayer bond distances in the collapsed-tetragonal (Ca,Sr,Ba)Cu2As2 compounds are much shorter than the nonbonding As-As distances in BaFe2As2. Thus the electronic character of the Cu and the strength of the As-As interlayer bonding are both expected to drastically change between weakly Cu-substituted BaFe2As2 and pure BaCu2As2, perhaps via a first-order lattice instability such as a miscibility gap in the Ba(Fe1−xCux)2As2 system.