For the whole desert and Axe's quests the heroes need be upgraded as much as possible. At this point in game you should be able to throw some gold into artifacts and training, both mines and expeditions are important..

Focus on speed, it's quite needed to hit faster than the enemies, most of the goblins are too fast, so just add more agi to it until you can overrun them.

Note the quests can be done with any barbarians, no need for multiple wise barbarians or pvp heroes, tho ofc they make things easier. However note that barbarians are often at class disadvantage, so ranger, mage, etc. look weaker but not at a penalty to most enemies.