Given that the Faith Points cap out at level +4, it looks like to me that this thing could spam as many as its level +4 extra attacks at the beginning of combat, and an extra 1-4 in every round after that.
There are plenty of ways to add extra damage, like traveling with a bard, multiclassing to get sneak attack (and taking Craven), or even just power attacking (since they're all at full BAB). And what's to keep this thing from not using its shield and power attacking with two hands?
I think this class feature should have a hard limit of one extra attack per round at lower levels, and maybe two at the upper ones. I didn't really look at the other jack classes, so I can't really comment on them. If' they're all like this, then I'd say that they all need to be changed.
I seem to have missed an important line about starting encounters with 0 faith points. I may also need a line about being unable to use Burning Wrath in two hands, no warjack has ever had a 2 handed weapon...
The ways to add extra damage are all equally applicable to a volley type archer (other than power attack) but I did run the expected damage output of the deathjack and it was within what I would consider a normal range at level 10.
Edit: Starting at 0 faith points added, made it impossible for the avatar to use two hands on its sword.
Maximum expected damage at Level 11:
6 attacks, 3d6(18)+(Str)+Power attack
Assuming starting 18 Str + 12 from class levels and 2 due to leveling up plus a +4 item gives 36 Str which is +13 damage. Assuming maximum power attack
6*(18+13+7)= 228 damage absolute maximum.
Level 11 Targeteer Fighter archer with Arrow Swarm, weapon specialisation and ranged weapon mastery wielding a Splitting Composite Longbow (Str cap +4)
3 attacks base, 1 extra from rapid shot, 2 extra from arrow swarm, doubled due to splitting.
12 attacks, 1d8(8)+4(Str)+2 (weapon specialisation) +2 (weapon mastery) =192 damage
Neither number is to be realistically expected in an actual encounter, both will generally only be a fraction of that.