The eidolon totally gets iterative attacks with a manufactured weapon when its base attack bonus is +6 or higher. From here...
Base Attack Bonus (BAB): Each creature has a base attack bonus and it represents its skill in combat. As a character gains levels or Hit Dice, his base attack bonus improves. When a creature's base attack bonus reaches +6, +11, or +16, he receives an additional attack in combat when he takes a full-attack action (which is one type of full-round action—see Combat).
...and here...
Base Attack Bonus: A base attack bonus is an attack roll bonus derived from character class and level or creature type and Hit Dice (or combination's thereof). Base attack bonuses increase at different rates for different character classes and creature types. A second attack is gained when a base attack bonus reaches +6, a third with a base attack bonus of +11 or higher, and a fourth with a base attack bonus of +16 or higher. Base attack bonuses gained from different sources, such as when a character is a multiclass character, stack.
Undoubtedly there are edge cases wherein a creature using a manufactured weapons doesn't get iterative attacks, but such a feature would be noted among the creature's abilities.
Natural weapons for the most part don't yield iterative attacks. Instead, the creature uses one natural weapon during a standard attack, or all its natural weapons on a full attack, with some natural weapons designated primary and other natural weapons designated secondary as per this chart; secondary natural weapon attacks suffer a -5 penalty to attack rolls and the damage bonus from the creature's Strength score is halved.
A creature wielding a manufactured weapon that also possesses natural weapons who takes the full attack action attacks with his manufactured weapon normally (including iterative attacks due to base attack bonus and usual damage bonus due to Strength) and can also make natural weapon attacks with any natural weapons not occupied with the manufactured weapon, but all such natural weapon attacks are considered secondary, suffering a -5 penalty to attack rolls and the damage bonus from the creature's Strength score is halved.
A cursory search should reveal questions probing the use of natural weapons in Pathfinder and its precursors and get answers that go into staggering detail.
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
The base form for the Biped Eidolon is:
So a base Eidolon will get two claw attacks with a FULL ROUND attack and one claw attack with a normal attack.
The "maximum of three attacks" is for when you start adding things like Bite and Sting. The Eidolon can only have a maximum of 3 attacks (at level 1) no matter how you augment it.
E.g. Tail, Bite and Sting are all 1 point evolutions; so if you took all those as your Eidolons evolutions at level one then it would have 4 attacks (Full round action) Claw, Claw, Bite, Sting. However the maximum number of attacks it is allowed is 3, so one of those attacks would be forfeit.
From your example
Each "attack" counts as an attack; so claw x2 and bite. Each one of those is an "attack", so that's three.
So for a FULL ATTACK the Eidolon would get three attacks (claw, claw, bite) For a standard action attack the Eidolon would get one attack, this would be either ONE claw or ONE bite as the Bite is listed as a PRIMARY attack - if you only have one attack then you select one of your primary attacks to use.