Step 1: Identify the polymers.
Reaction A is the anionic polymerization of styrene.
\[
X=\text{Polystyrene}
\]
Reaction B is the copolymerization of buta-1,3-diene and acrylonitrile.
\[
Y=\text{Buna-N (Nitrile rubber)}
\]
Step 2: Analyse the statements.
• Statement I: Polystyrene has stronger intermolecular forces than Buna-N due to the presence of bulky phenyl groups. Hence, this statement is correct.
• Statement II: Polystyrene is a thermoplastic polymer, not a synthetic rubber, whereas Buna-N is a synthetic rubber. Hence, this statement is incorrect.
• Statement III: Polystyrene is thermoplastic and Buna-N is resistant to oils and petrol. Hence, this statement is correct.
Step 3: Final conclusion.
Therefore, the correct statements are
\[
\boxed{\text{I and III only}.}
\]
Hence, the correct option is \(\boxed{(C)}\).