How do these all interact with each other? I have a player asking and I am not at a place where I can access the rulebooks at the moment(work). Want to see if there are already established answers before I go trying to start working it all out.
There has been a lot of discussion on unarmed options, and there's no single reading that everyone here ascribes to; Paizo has been noticeably quiet.
The basic impression is that Improved Unarmed Strike, Natural Weapons, and the Ring of Fangs all work together, though. Power armor unarmed is generally seen as its own thing, somewhat similar to Armor Storm's Hammer Fist ability.
There are two schools of thought on why the former three work; logical (assuming the natural attacks are as or more effective than regular weapons despite not specifically mentioning exceptions to existing rules) and textual (assuming that the only modifiers are explicitly stated in each item, and all other things remain equal to "normal" unarmed strikes):
Logical: While Natural Weapons benefits from the bonus damage from IUS, it doesn't need that feat to make attacks of opportunity or require a hand free, because the attacks are generally made with a different part of the body. Ring of Fangs is, effectively, a Natural Weapon because of how it is described.
Textual: Each of the three items or abilities adds a different set of modifiers to the Unarmed Strike weapon from the Equipment section. Natural Weapons adds potentially lethal, non-archaic damage, one-and-a-half specialization and the ability to threaten squares, IUS adds the ability to attack without a free hand and extra damage, and the ring of fangs adds (non-potentially) lethal damage and double specialization bonus. Individually, the options lack the benefits of the others (natural weapons need a free hand and "just" has 1.5x specialization, ring of fangs is archaic and needs a free hand, IUS is archaic, non-lethal, and has regular specialization).
I would suggest that you work out what seems best for your group; if the player in question is outshining everyone with his free natural weapons, consider the textual approach. If the party is only scraping by because the vesk can break out the claws at a desperate moment, go for the world-logic version.
Powered armor will set your strength score no matter what
If you have power armor that can attack and a deadly bite then you just have 2 different weapons ready. Not really that useful but you never know.
Imp unarmed strike only increases the die if you re wearing the ring of fangs [I think the ring removes archaic but that's a non issue with vesk anyway] the ring already did everything else.