1. I: In Leclanché cell, anode (Zn) oxidizes to Zn$^{2+}$; cathode reduces MnO$_2$ to Mn$_2$O$_3$ or MnO(OH), not Mn$^{2+}$ to Mn$^{3+}$, so incorrect.
2. II: Electrolysis of CuCl$_2$ with Pt electrodes: Anode oxidizes Cl$^-$ to Cl$_2$ (g) (2Cl$^-$ $\to$ Cl$_2$ + 2e$^-$), correct.
3. III: Lead storage battery is rechargeable (secondary battery), correct.
4. Thus, II and III are correct, so the answer is (3) II, III only.