The Mage Knights would have had to exist during the war, especially because Strago mentions something about how the Espers left them there, or 'after the Espers left' in reference to the humans hunting them. By the end of the war, a scapegoat was needed, and the Mage Knights made the perfect place to lay the blame.

And as for factions within the group, I'm sure there were all sorts on both sides, just as with every other conflict.