Step 1: Know what a dry cell is.
A dry cell is the common battery used in torches, clocks and remotes. It is a primary cell, which means it gives current by a one-way reaction and cannot be recharged.
Step 2: Check statement (I).
The dry cell is indeed the well-known Leclanche cell. So statement (I) is true.
Step 3: Check statement (II).
The electrolyte is a wet paste of ammonium chloride ($NH_4Cl$) and zinc chloride ($ZnCl_2$). This matches the real cell, so statement (II) is true.
Step 4: Check statement (III).
At the carbon (cathode) the reaction is $MnO_2 + NH_4^+ + e^- \rightarrow MnO(OH) + NH_3$. This is the correct cathode reaction, so statement (III) is true.
Step 5: Check statement (IV).
A dry cell is a primary cell. Its reaction is not easily reversed, so it cannot be recharged. Hence statement (IV) is false.
Step 6: Combine the results.
True statements are (I), (II) and (III). The false one is (IV).
Step 7: State the final answer.
The correct combination is:
\[ \boxed{\text{I, II, III only}} \]