Lots of movements and I need help understanding how this all would work in tandem.

[Finite_Fellow]

So this is a lot of feats to take into consideration but assuming you build a dual wielding fighter that has the following feats and all of the required prerequisites:

Greater feint
Spring attack
Power attack
Greater two weapon fighting

When hasted, could you feint an opponent, attack all 4 times and then spring step out in a single round? If not how many attacks could you get off and still be able to feint and spring step out ?

[Firebug]

TLDR: Basically, nothing except Power Attack works together with any of the other feats you listed.

First off, Haste no longer gives you an extra standard action if you were missing that change from 3rd edition D&D. It merely gives you an extra attack at your full attack bonus when you are making a full attack, a specific form of full-round action.

Likewise, Greater Two Weapon Fighting is only applicable during a full attack.

Feint is normally a Standard action. Greater Feint doesn't change the action type, but Improved Feint does, to a Move action. You have nothing in your build so far that gives you an extra move action (or allowing you to substitute an attack for a feint, or reducing it to a swift action/etc), so Feinting doesn't work with Full Attack. Or more specifically, it doesn't work with any Full-Round Action. Even Hasted.

Spring Attack is not a full attack, but it is a different full-round action. Improved and Greater Spring attack just add extra targets to your single attack. Specifically, the Attack Action, which is typically a standard action. It also does not give you a move action before and a move action after an attack (much less a full attack), it is just letting you split up your movement a little and negating Attacks of Opportunity from one target(and a 2nd and a 3rd with Improved and Greater).

Basically, nothing except Power Attack works together with any of the other feats you listed.

When Hasted, if you would feint, you would attack once. Done, no spring attack, no off-hand attacks, no iterative BAB attacks, no haste attack.

[Finite_Fellow]

Thank you that was a perfect explanation.

[Firebug]

You can get most of what you are looking for, just the whole full attack on spring attack is going to be rough. There are very few ways (not that I remember them off the top of my head) to get a full attack as a standard action. And even then, its not the 'attack' action so doesn't work with spring attack anyway.

But you can get some movement between attacks with Dimensional Dervish, Circling Mongoose or Jabbing Dancer.

Tack on the Two-weapon Feint or Moonlight Stalker to fit in the feint. Probably a few other ways to add it in as well, but those are the ones that come to mind.