For example: a multiclass Fighter 6 and Unchained Barbarian 6 makes a full-attack action against a target. In rage he has has +12 BAB, +6 STR, +2 attack bonus from rage and a +3 Greatsword. Would he attack with +23/+18/+13 with his Greatsword, +15 from Steel Headbutt and also +15 from Animal Fury?
Greatsword, headbutt, and bite during one full attack? Yes…
A barbarian 6/fighter 6 that's wearing medium or heavy armor that makes a full attack can make his manufactured weapon attacks normally then, because he's made a full attack, also use the extraordinary ability steel headbutt using his base attack bonus but at −5.
During that same full attack, such a warrior can employ any natural weapons he possesses—such as the bite granted by the rage power animal fury—as secondary natural attacks (therefore making each natural attack as if his base attack bonus were −5 lower and applying only half the warrior's Strength bonus on the the natural attack's damage roll).
A GM may limit a warrior's use of some natural attacks depending on the manufactured weapons he's employing during the full attack. For example, most GMs—including this one—rule it's impossible to use during the same full attack both a two-handed weapon and claws as the GM and the game often prejudiciously assume claws to be on hands instead of feet. Likewise, as Gnomejon mentions in this answer, a GM may rule a creature can't make a headbutt and a bite in the same round. (This GM, however, would allow a headbutt and a bite in the same round. It's not like a creature is doing both simultaneously. That is, a creature capable of multiple attacks with its greatsword isn't somehow making those attacks all at once. Combat is an abstraction, a turn representing an exchange of blows as well as parries and dodges and so on—not individual attacks. Were turns representative of actual events, everybody'd call it Slothfinder.) (For comparison, in GURPS events are actual instead of representative and turns are 1 second long.)
Anyway, more about natural attacks in Pathfinder can be learned from this question, this question, this question, this question, and more than a few others, too. (Apparently, Pathfinder should make this topic a little clearer.)
…But be careful with those attack bonuses
When a creature uses its base attack bonus −5 for something like a steel headbutt or secondary natural attack, it does not just get to use its highest attack bonus −5. Instead, the warrior subtracts 5 from his base attack bonus and computes his attack bonuses for those attacks separately and individually, adding bonuses and suffering penalties that apply only to those attack forms. For example, the enhancement bonus that applies to attack rolls with the +3 greatsword because of that weapon's +3 enhancement bonus don't count toward the attack bonuses of the steel headbutt and the bite.
(In other words, greatsword +3 at +23/+18/+13 and steel headbutt at +15 and bite at +15 is correct.)
There are no rules around using the same limb for different natural attacks, so it is available, RAW.
But as a GM
I'd say no, because it makes no sense that you are able to hit with both your forehead/top of the head and your mouth at the same time. Its one of those things that isn't defined well, but can be inferred by the following text in natural attacks:
Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (although often a creature must forgo one natural attack for each weapon clutched in that limb, be it a claw, tentacle, or slam).
Both Steel Headbutt and Animal Fury note that they require an additional -5 on top of not getting the +3 magical enhancement from the greatsword, so if you're doing a full attack with a weapon, they'd be at +15 to hit.