I don't know what Shoo will say, but the way I see it is:
The Holy Swordsman imbues his sword with his soul when he strikes down his enemies.
It could explain why the blade changes color when one of their abilities is used.
I'm a little confuse as to what you mean by driving mentality but I'll try the best I can. I would think that their mentality would be to fulfill their gods wishes. ATM they are an army made to destroy the demon. So they would always work towards that goal. As to how they would act, that's just like any other class/race/player. They could all be completely different in how they do things.
"Are they just fighters on a crusade to slay 'The Demon Krovak'? If so, what happens if a Holy Swordsman slays him? Once the others find out, what is their purpose after that?"
They are an army that specializes in this style of fighting, but they have the capability to use other forms as well. So, no they are not just fighters on a crusade, they are a force of their god on a crusade. If a Holy Swordsman slays Krovak then their current mission is complete. As we all know about gods, they live to grant quests for those who worship them. When Krovak is destroyed, the Sky God and all of his people will turn to Bahamut for their next challenge. Until he grants them the privilege of his presence, they will live their lives as any other nation.