I like the concept, though I would like to see it in play to see how it would balance out. The Axe's bonus Regen feels pretty high, and you state explicitly that the ammo of the weapons isn't increased by %ammo effects (which makes sense considering these are more abilities than actual guns).
I'm also not clear on how his Staff and the Axe are supposed to work together. Yes it gives him move speed and evasion, but why does it have knockback? That feels like it would work directly against a melee weapon like the axe or the Shotgun effect of his 1. Maybe the energized effect of the axe causes his shots with it to leap forward a bit?
Also, while his 1 having Knockback seems pretty cool, again I'm not seeing a lot of cohesion between his abilities. It's clear he's supposed to be an up close brawler type, but his weapons seem to have effects that work against that.
For example, Haze's Sleep dagger functionally stops an enemy who might be out of position, allowing haze to rush in and hit her ult on them. Her Fixation increases damage on enemies shot (which is what her ult does) and her stealth helps her get into a good place to hit the skillshot of her dagger or get into the middle of the enemy team to ult.
What is an ideal combo for his abilities? How are they supposed to work together? Right now, it feels like some nifty concepts all thrown onto 1 character. The individual ideas are great, but they need to work together in some way, or the hero won't be much fun to play
You also might want to consider making it so that gaining bonus charges affects ammo capacity instead, though. Alternatively, the ultimate could have charges (which indirectly also increases the ammo of your special weapons).
Looking at it as well, I can't think of any situation where you wouldn't max the Ult first. It has impact on all of your other abilities, so there's just no reason not to. Maybe the special enhanced effects should be either lower on the ability tree? Like the 2 point effect? Not sure. It would need to see play, but my gut says every build will go Ult max first.
And if I understand correctly, you're supposed to use his ultimate after you equip a weapon? Does that mean that his base gun would also have an ultimate effect? I think that would be a great way to make him have both solid Spirit and Gun paths.