You almost always need a Full-round Action to take multiple attacks
Natural Weapons are not an exception
As a standard action, you can use any of your natural weapons to make a single attack. You cannot make attacks with multiple natural weapons, not even paired weapons like claws. If a weapon is “secondary” (it will say so in its description), it makes the attack at −5 even if it’s your only attack.
As a full-round action, you may attack once with each weapon. Only one weapon or pair of weapons can be “primary;” even if you have more than one weapon or pair of weapons that claim to be “primary” in their description, you must pick one or one pair to be primary, and all others are secondary. Secondary natural weapon attacks are taken at a −5 penalty.
This means your attack routine with two claws (primary pair) and a tail attack (secondary) is +0/+0/−5 at BAB +0.
There are some (rare) ways to attack more than once as a standard action, and some of them are compatible with natural weapons. These will explicitly state this exception to the usual, however.
Natural Weapons and Manufactured Weapons use totally separate rules
Natural weapons never receive iterative attacks, and there are no cumulative −5 penalties, only a single −5 penalty applied to all secondary natural attacks. If you have the Multiattack feat, all secondary natural weapons attack at a −2 penalty instead of a −5 penalty.
None of the rules about manufactured weapon attacks (iteratives, two-weapon fighting, Flurry of Blows, etc.) apply to natural weapons.
Thus at BAB +20, that same pair of primary claws and secondary tail slap attacks at +20/+20/+15, or +20/+20/+18 with Multiattack. If you added a pair of secondary Tentacle attacks, that would be +20/+20/+15/+15/+15 without Multiattack, or +20/+20/+18/+18/+18 with.
Unless otherwise specified by the natural weapon, primary natural weapons add your Strength modifier to damage, while secondary natural weapons add half your Strength modifier to damage.
You can mix manufactured weapons and natural weapons.
When you do so, you gain all your normal manufactured-weapon attacks (including iteratives), plus any natural weapons you have, each as a secondary weapon, with the caveat that you cannot use any natural weapons that are being use to hold a manufactured weapon. I.e. if you hold your sword in one hand and attack with it, you cannot use the Claw attack from that hand. If you instead two-hand a Great Axe, then you cannot use either Claw attack.
When you do this, you can mix natural weapons with full-attacks that use the two-weapon fighting combat option or Flurry of Blows class feature.
Example: Longsword, pair of claws, and tail slap
Thus if you had a Longsword, a pair of Claws, and a Tail Slap, your attack routine would be +0/−5/−5 (sword, claw, tail) for BAB +0. Multiattack improves this to +0/−2/−2. At BAB +20, it’s instead +20/+15/+10/+5/+15/+15 (sword, sword, sword, sword, claw, tail) or +20/+15/+10/+5/+18/+18 if you have Multiattack.
Same character with dual kukris and the Two-Weapon Fighting feat
If you instead have the Two-Weapon Fighting feat, and use that option with a pair of light weapons (call it a pair of Kukris), you attack at −2/−2/−7 (kukri, kukri, tail), since both hands with Claws are being used to attack with Kukris, and the Tail is always secondary when used with manufactured weapons (hence −5; note that Two-Weapon Fighting does not penalize it, however). The Multiattack feat would reduce the penalty, so the tail would be at −2 (+3 attack for one out of three attacks is not a great use of a feat, but just for completion’s sake).
At BAB +15 and with the Improved and Greater Two-Weapon Fighting feats, you get +13/+13/+8/+8/+3/+3/+10 (left kukri, right kukri, left kukri (2nd), right kukri (2nd), left kukri (3rd), right kukri (3rd), tail).
Replacing a claw attack with an unarmed strike
Sorry, but no. Unarmed strikes use a weird hybrid of the rules for manufactured and natural weapons, but for the purposes of full-attacks, they work like manufactured weapons. That is, you get iteratives with them, but if you can only combine them with natural weapons by making those natural weapons secondary (−5 attack penalty, only ½Str to damage).
The first rule that you quote is specifically about spells and effects. A full-attack is not either of those.
Claws and lack of offhand unarmed strikes
Feral Combat Training does mean that anything from the monk’s unarmed-strike-improving class features can apply to natural weapons, and that can include the bit about never being offhand.
However, claws and other natural weapons are never “offhand” to begin with. The term “offhand” only applies when using two-weapon fighting, and that combat option does not interact with natural weapons (aside from the attack penalty, which applies to all attacks). So the fact that the monk class feature, combined with Feral Combat Training, says that natural weapons are never offhand does not do anything because that was already true.
Instead of “main hand” and “offhand,” natural weapons are either “primary” or “secondary.” These are different. When combined with manufactured weapons (or unarmed strikes) in a given full-attack, all natural weapons are secondary: they receive the −5 penalty and get only ½Str to damage. Neither the monk class nor Feral Combat Training does anything about treating them as secondary or removing or reducing the penalties for being secondary.
So whether you have Feral Combat Training or not, your full-attack using unarmed strikes is:
Unarmed Strike, Claw (−5), Claw (−5), Claw (−5)
If you have Feral Combat Training, the claws do benefit from the improved base damage dice of unarmed strike, however, even if they’re still stuck with ½Str to damage.
Two-Weapon Fighting, Feral Combat Training
If you are actually using two-weapon fighting, the provision about monks never having offhand unarmed strikes meaningfully applies only to the unarmed strike. It “applies” to the claws, but does nothing for them.
So, for example, if your two weapons are a sai and an unarmed strike, and you have the Two-Weapon Fighting feat, your attack routine would be:
Sai (−2), unarmed strike (−2), claw (−5), claw (−5)
The unarmed strike would add your full Strength to its damage, however. Note that I assumed that the sai took up one of your claw-hands. I did not wish to get into the debate about whether one can use two unarmed strikes as part of two-weapon fighting.
You didn’t ask, but about Flurry of Blows
All of the statements above about full-attack apply equally well to flurry of blows, except that you need Feral Combat Training to use natural weapons in a flurry at all, and flurry of blows cannot be combined with two-weapon fighting because of Paizo nonsense.
Personal recommendation
For the record, monks, natural attacks, and how they combine, these are some of the worst things in Pathfinder. The rules are confusing, complicated, and the result works very poorly. I suggest you save yourself a headache and just... not.
Best Answer
What is a Weapon
This question really boils down to this: Are Natural Attacks considered Weapons?
Technically there is no official rules text stating that natural attacks are or aren't weapons, but fortunately we have this FAQ from the Pathfinder design team:
So yes, natural attacks are weapons of the light category.
Heart of The Metal
So on to your actual question. The Targets property of the spell designates "one weapon per level". In the text of the spell, it specifically states "This is able to affect nonmetal weapons." Thus, the spell is able to target any weapon.
Natural attacks are weapons, and Heart of The Metal can target any weapon, so yes, you can target your natural attacks with this spell. Just make sure you have the material component on hand!