You can wield & attack with two weapons without any special class feature or feat.
Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):
Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.
Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.
The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.
Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.
No, there are no such rules for sunder in particular
Excepting special cases for particular monsters, as with hydras or krakens.
The rules for sunder explicitly begin with
Sunder
You can use a melee attack with a slashing or bludgeoning weapon to strike a weapon or shield that your opponent is holding.
(emphasis mine)
There is an additional section for a carrier or worn object, but that still doesn’t apply to natural weapons.
In online discussions of sunder, the inability to use it on natural-weapon-using monsters is frequently brought up as a massive problem with the tactic (though far from the only one). I have never seen anyone mention any official rule for sundering them, nor have I ever seen such a rule myself, strongly supporting my broader claim that no such rule was published in a supplement. I specifically checked Rules Compendium, as the most likely location of such a thing, and it only reprints what core had to say (plus a little sidebar on how a DM should prepare encounters when a PC is using sunder).
As for houserule, I have not used, or seen used, any particular rules. Most seem willing to accept that sundering is just a dead rule, that is undesirable for PCs and (often) obnoxious and unfun for use by NPCs. There have been few, if any, attempts to rehabilitate it, and I’m not familiar with any. Your approach of using the hydra’s rules seems appropriate, though I wouldn’t guess that bone and hide, even fantastic bone and hide, has the same Hardness as steel.
There may be optional, variant rules for called shots, but....
I have to admit that I am not familiar with any official variant rules for called shots, but I suspect they exist (and homebrew versions certainly exist as well). However, I would caution against them: they are, in effect, like super-charged criticals. Many even run off of critical mechanics, but even when they don’t, by definition you are talking about something with a lower chance of a higher consequence.
The problem with this is that it degrades the stability of the system. The ability to predict consequences and prepare for them is diminished, and the game was already quite swingy. I would argue, then, that these effects are to the detriment of the game.
You may disagree; you might want something even swingier. That’s fine, as long as you have a group that’s on board with that, but I think it is important that you and your group all know the ramifications here: swinginess is inherently bad for the players. Mathematically, a dire consequence as a result is equally likely for PCs and NPCs (assuming they’re making similar numbers of attempts, which seems mostly reasonable), but while such bad luck against NPCs is much more likely than not to befall some random mook, the same bad luck against PCs will land on a PC 100% of the time by definition. And since there are relatively low risks of these consequences, it is more difficult to defend against them—player resources are already devoted to a number of things they need to defend against as it is, so it will be difficult to justify diverting some of them to protect against a low-risk event.
Called shots are far better than critical or fumble tables in that they are specific, planned maneuvers, so they avoid a lot of the huge narrative failings of critical or fumble tables. So that much is good. And since you are presumably giving up an attack or something else to attempt these called shots, your odds can be reasonable, and if the odds are reasonable, the effects need not be so dire. So a well-made system could mitigate some of the mathematical problems I mention above. But it is important that, whatever you choose, you go into it with eyes open, aware of potential problems. Unlike critical or fumble tables, they’re not insurmountable for a group that’s interested in this sort of thing, but they’re not something to add on a whim because it sounds cool. Consider it carefully.
Best Answer
You are not missing anything
A second attack with your claws is already baked into the Claws feature, it just does not cost you a bonus action as two-weapon fighting would. The most natural way to think of this is that you attack with both claws, but technically, you do not:
This does not require you to have two free hands/claws, and could be done with a single free claw (for example, if you hold a shield in the other).
Two-weapon fighting normally (without feats) requires light melee weapons in both hands. All of those in the core rules do d6 or less damage. That means, there will be no improvement for the off-hand claw attack, and no improvement for the first main-hand claw attack, but as you say, you get another main hand claw attack for free compared to normal two-weapon fighting from the extra claw attack of Claws.
This certainly is a tangible power-up compared to the rules as written, but at +3.65 damage before to hit chance per round it won't break the game in a way that is significantly unbalancing.
Especially not as you have access to extra attack and feats already. Two-weapon fighting is generally seen as one of the weaker melee combat options that at higher levels loses steam compared to polearms or great weapon fighting with the appropriate feats, even when you have the Dual Wielder feat. And here you cannot use that effectively as it allows you to use larger weapons, and you don't use weapons. So the homebrew rule would help to offset a weaker option to make it compare better and be more attractive.
The only possible risk I can think of would be with features that pay off on a critical hit (for example, if the Barbarian also would take Paladin levels for Divine Smite), as more attacks means more chances to land that hit. But I do not think that this would be more abusive than other optimized builds, if at all.