Going down the list, adding additional items/attacks as we go to see how the attack sequence is built up.
Horned helm
The horned helm in Magic Item Compendium gives a gore attack, which is a natural weapon. Natural weapons use completely separate rules from weapon attacks, but can be combined with those by treating natural weapons as secondary (−5 on attack, ½Str bonus on damage). Other than this, there is no interaction, you just make this secondary natural weapon attack alongside your weapon attacks. See this Q&A for more details.
Horned helm + two-hander
Most relevantly to this question, a gore attack doesn’t interact with two-weapon or multiweapon fighting; the gore attack is neither your primary hand nor your offhand, so none of the penalties from those fighting modes apply.
These secondary attacks do not interfere with the primary attack as attacking with an off-hand weapon does,
(SRD → Special Abilities → Manufactured Weapons)
It also doesn’t count as a second (or third or whatever) weapon for the purposes of whether or not to use those modes; if you used a two-handed weapon and then also used your gore attack, you would not be using two-weapon fighting and would not take those penalties or use any related feats.
Horned helm + two-hander + armor spikes
Armor spikes are an offhand weapon that don’t actually require a hand. You can use two-weapon fighting with a two-handed weapon and armor spikes, and you would take the “off-hand weapon is light” penalties listed here. If you have the Two-Weapon Fighting feat (and you should if you are doing this), that means you take a −2 penalty on the two-hander as well as a −2 penalty on the armor spikes. You could add the horned helm’s gore attack to this as a secondary weapon; it would take a −5 penalty for being a secondary natural weapon, but would not take any penalties related to the two-weapon fighting you’re doing with the two-hander and the armor spikes. It also wouldn’t affect those penalties in any way.
Horned helm + two-hander + armor spikes + unarmed strike
You can’t actually do this, interestingly enough:
(You can’t also make an attack with armor spikes if you have already made an attack with another off-hand weapon, and vice versa.)
(SRD → Weapons → Armor Spikes)
Anyway, even without this line, unlike natural weapons, manufactured weapons require you to have some “attack” available to use the weapon with. Just having the weapon doesn’t allow you to use it. (In contrast, just having a natural weapon does allow you to use that.) That’s why you need two-weapon fighting in the first place: you could be standing there with a pair of weapons, but ordinarily you’d only get to attack with one of them. You need a second attack in order to attack with the other one. That second attack could come from haste or BAB +6 or whatever, but those attacks could be made with the same weapon as the first, so it’s usually better to just keep using one weapon (and make that weapon bigger and better). Two-weapon fighting, on the other hand, is available at lower levels than haste or BAB +6, and stacks with those, allowing you to get another attack—but with the requirement that the second attack be made with a different weapon from the first. So you don’t get an extra attack just for having a second weapon, instead you use two-weapon fighting which lets you make another attack with a second weapon.
So you could not use two-weapon fighting to get offhand attacks with some light weapon (other than armor spikes) and then also with an unarmed strike. Two-weapon fighting only allows you one extra attack, so you have to pick which weapon to take it with.
“But wait, what about multiweapon fighting?”
I hear you ask. Well, I wish you wouldn’t.
The full and complete rules we have for multiweapon fighting are these:
Normal A creature without this feat takes a -6 penalty on attacks made with its primary hand and a -10 penalty on attacks made with its off hands. (It has one primary hand, and all the others are off hands.) See Two-Weapon Fighting.
What is this “normal” entry here? It’s a section of the rules for the Multiweapon Fighting feat. They describe a “normal” scenario that isn’t described anywhere else. It instructs us to read up on two-weapon fighting, but the two-weapon fighting rules are all about two weapons, and don’t make any note of having more than that.
Multiweapon Fighting itself has a “three or more hands” prerequisite. If you don’t have that, you can’t take Multiweapon Fighting, so we don’t have to worry about it.
Unfortunately, the “normal” section is supposed to be describing—really, reminding—what the deal is without the feat. That means even without taking it, we have to worry about what this means. (Also, if you have three or more hands, you could still ask what happens if you use weapons in all of them and then also kick someone with an unarmed strike.)
And at that point, I have to just throw my hands up in the air and shrug. We just don’t know what the rules of this hypothetical multiweapon fighting special attack option are, because they didn’t actually write any. The existence of this fighting style is implied by the Multiweapon Fighting feat, and we have examples of that feat in use and creatures using more than two weapons in a full-attack (most notably the marilith’s six arms with six weapons), but we don’t know where the edges, limits, and corners of these rules are. I don’t know if you could throw in an unarmed strike or not. I don’t know if you can do that even if you don’t have three or more hands. I don’t know how creatures that wield weapons in non-hand appendages work. I certainly don’t know what “[Multiweapon Fighting] replaces the Two-Weapon Fighting feat for creatures with more than two arms” means.
The fact that the rules don’t cover this, and tossing in extra unarmed strikes is not considered or suggested anywhere in the books, suggests that the answer is that you can’t, at least not without three or more hands. Beyond that, we’re left to speculate.
