A series of new heteroleptic copper(II) complexes of the composition [Cu(L)(bpy)]NO3·2MeOH (1), [Cu(L)(dimebpy)]NO3·2H2O (2), [Cu(L)(phen)]NO3·2MeOH (3), [Cu(L)(bphen)]NO3·MeOH (4), [Cu(L)(dppz)]NO3·MeOH (5) was prepared, where HL = 3-(3,4-dihydroxyphenyl)-5-hydroxy-8,8-dimethyl-6-(3-methylbut-2-ene-1-yl)-4H,8H-benzo[1,2-b:3,4-b']dipyran-4-one, (pomiferin) and bpy = 2,2'-bipyridine, dimebpy = 4,4ʹ-dimethyl-2,2ʹ-bipyridine, phen = 1,10-phenanthroline, bphen = 4,7-diphenyl-1,10-phenanthroline, and dppz = dipyrido[3,2-a:2',3'-c]phenazine. The complexes were characterized using elemental analysis, infrared and UV/Vis spectroscopies, mass spectrometry, thermal analysis and conductivity measurements. The in vitro cytotoxicity, screened against eight human cancer cell lines (breast adenocarcinoma (MCF-7), osteosarcoma (HOS), lung adenocarcinoma (A549), prostate adenocarcinoma (PC-3), ovarian carcinoma (A2780), cisplatin-resistant ovarian carcinoma (A2780R), colorectal adenocarcinoma (Caco-2) and monocytic leukemia (THP-1), revealed the complexes as effective antiproliferative agents, with the IC50 values of 2.2–13.0 μM for the best performing complexes 3 and 5. All the complexes 1–5 showed the best activity against the A2780R cells (IC50 = 2.2–6.6 μM), and moreover, the complexes demonstrated relatively low toxicity on healthy human hepatocytes, with IC50 > 100 μM. The complexes were evaluated by the Annexin V/propidium iodide apoptosis assay, induction of cell cycle modifications in A2780 cells, production of reactive oxygen species (ROS), perturbation of mitochondrial membrane potential, inhibition of apoptosis and inflammation-related signaling pathways (NF-κB/AP-1 activity, NF-κB translocation, TNF-α secretion), and tested for nuclease mimicking activity. The obtained results revealed the corresponding complexes to be effective antiproliferative and anti-inflammatory agents.