Only your class quests will award you with gladiator specific reqards. The main story quests and sidequests always let you pick what you want, typically offering you a choice of one DoM item, one DoW-dps item, one DoW-tank item or some gold. So you won't be missing out on anything by doing the main quests in any class you choose.

Also, bear in mind that the best gear will not be obtained by completing quests. The best loot always comes from the dungeons.