Arene–ruthenium(II) complexes with carbothioamidopyrazoles at the C-2 and C-5 positions have been recognized as chemotherapeutic agent alternatives to cisplatin and its oxaliplatin analogs. The aim of this study was to continue research on the biological aspect of arene–ruthenium(II) complexes and their anticancer activity. The present paper includes an additional 12 new tumor cells, analyzed by MTT, and employs a series of extended bioassays to better understand their potential mechanism of antitumor activity. The following tests were conducted: membrane permeability studies, intramolecular reactive oxygen and nitrogen species (ROS/RNS) assays, mitochondrial potential changes, DNA analysis by comet assay using the electrophoresis method, measurement of cleaved PARP protein levels, and determination of apoptotic and necrotic cell fractions by fluorescence microscopy. Additionally, the article presents lipophilicity studies based on RP-TLC and molecular docking studies. We hope that the presented data will prove useful in practical treatment, especially for patients with cancer.