To start with, heroism isn't defined whether you kill or not, like I said there's plenty of characters given as heroic who will readily kill threats because of a lack of social constructs that make such a thing unnecessary like effective law enforcement and prisons to keep people safe without resorting to taking away a life. I'm not particularly seeing the problem with a guy who clears out dens of dangerous creatures for the peasantry, he'd probably be hailed as a hero for getting rid of a threat to everyone's safety.

Distaste for magic, religion and nobility doesn't mean he won't work with them, but he sure as hell won't be impressed by magic, religious devotion or noble birth. He sees magic as taking a crappy shortcut to power rather than building oneself up and relying on their own body, religion as hiding behind something you feel is greater and waiting for things to be given for granted rather than actually striving towards them and noble birth as completely meaningless, because you are much the same as anyone else.

The strong rule the weak in Ajax's world. Be it by establishing an order that puts them ahead by use of state or religion, it winds up with some people on top and some people on the bottom. Ajax feels that everyone should strive to be higher than they are, to be strong. Not many can walk the same path he has, to be physically strong to such a level that it lets them do as they will, and he'll certainly look down on those who don't as inferior to him, but any who can place themselves in a position of power over others should remain there.

This applies in every way, of course. If those people who put themselves over others are then, in turn, overthrown and others establish their own order in some way then that is the way of life. He sees his path as the best, much like most everyone does theirs, but this doesn't necessarily make him aggressive towards those who don't follow it.

As far as party dynamics, think of him as that one buddy of yours who doesn't necessarily agree with what you do because you're an art major and he's a lawyer or a surgeon and he teases you gives you crap about it but doesn't particularly care and can live with you being wrong. See what I did there?