Option 1 -> Incorrectly matches "flog a dead horse" with wanting a superior post for personal gain.
Option 2 -> Incorrectly matches "add fuel to fire" with seeking a superior post, and misplaces other idioms.
Option 3 -> Incorrectly matches "cut no ice" with the arrest situation and "feather his own nest" with the guilty plea.
Option 4 -> (A)-(III): "feather his own nest" = use position for personal gain; (B)-(I): "add fuel to the fire" = worsen the situation; (C)-(IV): "flog a dead horse" = waste effort on hopeless task; (D)-(II): "cut no ice" = have no effect.
Hence, Option 4: (A) - (III), (B) - (I), (C) - (IV), (D) - (II) -> The idioms perfectly match the contexts: seeking personal enrichment from power, worsening an already tense situation, attempting something futile with a miser, and appeals that fail to persuade -> correct