So if I have the Panther style feat chain (in its entirety)and shift into let's say a bear, would I be able to use these feats while being a bear?
There aren't any rules for this. Once again, Paizo's humancentric rule writing rears its ugly head. RAW, this works in your favor, because there are rules that allow any PC to make unarmed strikes, and no rule that says you lose them while polymorphed.It is, however, intended that creatures without humanoid form don't have unarmed strikes, as indicated by this part in the Bestiary: "Some fey, humanoids, monstrous humanoids, and outsiders do not possess natural attacks. These creatures can make unarmed strikes, but treat them as weapons for the purpose of determining attack bonuses, and they must use the two-weapon fighting rules when making attacks with both hands."
Be sure to clear this up with your GM, because the last thing you want is to snub them - Panther Style stops working the second the GM wants it to, because making an AoO is optional, not mandatory.
This is always a problem with Panther Style, and the extremely gamey mechanic can be enough to annoy a GM, but you have to be extra careful, lest you end up having any at least intelligent enemy stop making those AoO when you run around them in circles.
Addition question, if I took the Fiend totem rage powers would they also function with this in beast form?
This, however, has a rule, one that explicitly enables it: "You also lose any class features that depend upon form, but those that allow you to add features (such as sorcerers that can grow claws) still function." CRB pg. 183
Ok, so because of the wording on Beast shape (which is how the archtype turns me into an animal) it doesn't actually give me any natural attacks, which means Panther style should still function normally. There is the point that you pointed out about Panther style, but it shouldn't be too bad of a problem for the current game I'm in. We go into a dungeon that has litterally thousands of monsters all attempting to just slaughter people that step in so I imagine the AoO shouldn't be too big of a problem.
Ok, so because of the wording on Beast shape (which is how the archtype turns me into an animal) it doesn't actually give me any natural attacks, which means Panther style should still function normally.
You always get the natural attacks of the form you polymorph into. "In addition to these benefits [from the spell description], you gain any of the natural attacks of the base creature, including proficiency in those attacks." CRB pg. 212 For a bear, 2 claws + bite.