You are considered to have the two weapon fighting feat when using a flurry of blows maneuver with some exceptions. When using a flurry of blows, all attacks under the flurry are at full strength modifier and your base attack bonus in monk levels is replaced with your level in monk (3 BAB from monk becomes 4 BAB). Normally a monk would hit level 8 and not need Improved TWF, or hit 15 and not need Greater, but because you are level 4, you can use the Improved and Greater.
This means that when you use a flurry of blows, you should have a BAB of 16 -> 4 attacks. In addition, because you only have 4 levels of monk, Improved TWF and Greater TWF will stack however they will only use .5 str (I am second guessing myself on this one, I think you still get to apply full strength because of flurry)
BAB +16(-2) / FOB +16 (-2) / BAB +11(-2) / Improved TWF +11(-2) / BAB +6(-2) / Greater TWF +6(-2) / BAB +1(-2)
A monk cannot use any weapon other than an unarmed strike or a special monk weapon as part of a flurry of blows. A monk with natural weapons cannot use such weapons as part of a flurry of blows, nor can he make natural attacks in addition to his flurry of blows attacks.
To answer your second question you have Claw/Claw/Bite = 3 natural attacks for the multi attack feat.
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.
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.