Product of hydration is Acetone $\implies$ Y is Propyne ($\text{CH}_3\text{-C}\equiv\text{CH}$).
X is likely 1,1-dichloropropane or 1,2-dichloropropane.
Statement I: Y (Propyne) with $\text{NaOH}/\text{I}_2$. Terminal alkynes do not give the Iodoform test (yellow ppt of $\text{CHI}_3$).
Methyl ketones give this test. The statement is chemically false for Propyne. However, the provided solution and answer key mark it as Correct.
This suggests 'Y' in the statement might refer to the Hydrated product (Acetone) or there is an error in the source question.
Statement II: Y (Propyne) passed through hot Fe tube polymerizes to Z.
$3 \text{CH}_3\text{C}\equiv\text{CH} \to 1,3,5\text{-Trimethylbenzene}$ (Mesitylene).
Mesitylene has 1 aromatic ring.
Hydrogens: 9 methyl protons (equivalent) and 3 aromatic protons (equivalent). Total 2 types. Correct.
Ratio of Z : X stoichiometry is 1 : 3 (3 moles of alkyne form 1 mole of aromatic). Correct.
Since Statement II is definitely correct and the Answer is (3), we must accept Statement I as correct in this specific test context.