The correct statement(s) regarding the periodic properties of...

For (A): The electronic configuration of $\text{B}^+$ is $1s^2 2s^2$ and that of $\text{C}^+$ is $1s^2 2s^2 2p^1$. Removing an electron from the fully filled, more penetrating $2s$ orbital of $\text{B}^+$ requires more energy than removing a $2p$ electron from $\text{C}^+$. Thus, the second ionization enthalpy of carbon is less than that of boron. Statement (A) is correct.

For (B): $\text{Al}^{3+}$, $\text{Mg}^{2+}$, and $\text{Na}^+$ are isoelectronic species with $10$ electrons each. The nuclear charge increases in the order $\text{Na}^+ (11) < \text{Mg}^{2+} (12) < \text{Al}^{3+} (13)$. Higher effective nuclear charge leads to a stronger pull on the electrons, resulting in a smaller ionic radius. Thus, the order of ionic radii is $\text{Al}^{3+} < \text{Mg}^{2+} < \text{Na}^+$. Statement (B) is correct.

For (C): The density of potassium ($0.86 \text{ g/cm}^3$) is less than that of sodium ($0.97 \text{ g/cm}^3$) due to an abnormal increase in atomic volume caused by the presence of empty $3d$ orbitals in potassium. Statement (C) is incorrect.

For (D): The bond dissociation enthalpy of $\text{H}-\text{H}$ ($436 \text{ kJ/mol}$) is much higher than that of $\text{F}-\text{F}$ ($159 \text{ kJ/mol}$) due to strong interelectronic repulsions between the lone pairs on the small fluorine atoms. Statement (D) is incorrect.

Answer: Second ionization enthalpy of carbon atom is less than that of boron atom.; Increasing order of ionic radii: $\text{Al}^{3+} < \text{Mg}^{2+} < \text{Na}^+$