1. Capacitance ($C$): The capacitance of a parallel plate capacitor is given by $C = \frac{\varepsilon_0 A}{d}$. When plates are moved closer, the distance $d$ decreases, causing capacitance $C$ to increase. (Statement (C) is Correct).
2. Potential ($V$): Since the capacitor remains connected to the battery, the potential difference $V$ across the plates remains constant.
3. Charge ($Q$): Charge is given by $Q = CV$. Since $C$ increases and $V$ is constant, the charge $Q$ increases. (Statement (A) is Correct).
4. Energy ($U$): Energy stored is $U = \frac{1}{2}CV^2$. Since $C$ increases and $V$ is constant, the energy $U$ increases. (Statement (B) is Incorrect).
5. Ratio ($Q/V$): The ratio $Q/V$ is equal to capacitance $C$. Since $C$ increases, the ratio increases. (Statement (D) is Incorrect).
6. Product ($QV$): The product $Q \times V$ represents twice the energy ($2U$) or the work done. Since $Q$ increases and $V$ is constant, the product $QV$ increases. (Statement (E) is Correct).

Conclusion: Statements (A), (C), and (E) are correct.