Why does an ionic compound have to be molten to undergo electrolysis?

When lead iodide, ${\mathrm{PbI}}_2$ is electrolysed, the following reactions occur:
${\mathrm{Pb}}^{2+} + 2{\mathrm{e}}^{-} \to \mathrm{Pb}$ and
$2{\mathrm{I}}^{-} \to {\mathrm{I}}_2 +2{\mathrm{e}}^{-}$

Which of these equations describes the reaction at the cathode? Explain your answer.

State the oxidation number change of each iodide ion in the reaction: $2{\mathrm{I}}^{-} \to {\mathrm{I}}_2 + 2{\mathrm{e}}^{-}$

Calculate the volume of hydrogen produced at r.t.p. when a concentrated aqueous solution of sulfuric acid is electrolysed for $15.0 \min$ using a current of $1.40 \mathrm{A}$. ($\mathrm{F}= 96500 \mathrm{C} {\mathrm{mol}}^{-1}$; $1 \mathrm{mole}$ of gas occupies $24.0 {\mathrm{dm}}^3$ at r.t.p.) (Write the value only rounded off to two decimal places)

A learner passed a constant electric current of $0.15 \mathrm{A}$ through a solution of silver nitrate, using pure silver electrodes, for $45 \min$ exactly. The mass of the anode decreased by $0.45 \mathrm{g}$. Use this data to calculate the charge on a mole of electrons. (A_r value: $\left[\mathrm{Ag}\right] = 108$)