In the previous game, you get each class strongest weapon each time you defeat a boss in the extra dungeon after beating the game. I think it's same for this game.
Not really, I've already written the information concerning the Strongest Weapon. In the Extra Dungeon you're require to defeat the Dragon on the map (Purple Icon) those are random drop so equip the accessory which increase your drop rate. To respawn them simply Exit the dungeon and re-enter.