This study examined the antiradical activity of three synthesized coumarin derivatives: (E)-3-(1-((2-hydroxyphenyl)amino)ethylidene)-2,4-dioxochroman-7-yl acetate (A1-OH), (E)-3-(1-((3-hydroxyphenyl)amino)ethylidene)-2,4-dioxochroman-7-yl acetate (A2-OH), and (E)-3-(1-((4-hydroxyphenyl)amino)ethylidene)-2,4-dioxochroman-7-yl acetate (A3-OH) against HOO•/O2•− radical species. The investigation included electron spin resonance (ESR) measurements and a DFT kinetic study. Thermodynamic and kinetic parameters of antiradical mechanisms—Formal Hydrogen Atom Transfer (f-HAT), Radical Adduct Formation (RAF), Sequential Proton Loss followed by Electron Transfer (SPLET), and Single-Electron Transfer followed by Proton Transfer (SET-PT)—were evaluated using the Quantum Mechanics-based test for Overall Free Radical Scavenging Activity (QM–ORSA) under physiological conditions. ESR results indicated antiradical activity decreased in the sequence A1–OH (58.7%) > A2–OH (57.5%) > A3–OH (53.1%). Kinetic analysis revealed the f-HAT mechanism dominated HOO• inactivation. A newly formulated Sequential Proton Loss followed by Radical Adduct Formation (SPL-RAF) mechanism described interactions with O2 •−. The activity toward O2 •− was A2–OH (1.26 × 106 M−1s−1) > A3–OH (7.71 × 105 M−1s−1) > A1–OH (4.22 × 105 M−1s−1). Molecular docking and dynamics studies tested inhibitory capability against enzymes producing reactive species: Lipoxygenase (LOX), Myeloperoxidase (MPO), NAD(P)H oxidase (NOX), and Xanthine Oxidase (XOD). Affinity to enzymes decreased in the order: XOD > LOX > NOX > MPO.