If you take monk you can either use your natural weapons or your flurry. If you take the feral combat training you can make your flurry attacks with your natural weapons. But as far as I know there is no option of making both the flurry attacks and the natural attacks.
When you rage you get the increased stats, not your eidolon. So I do not think you benefit in any way from it.
Which means that since without feral combat training flurry as an eidolon isn't good, and if your eidolon wont get the bonus from rage that leaves the option of natural weapon archetype fighter being probably the best choice.
Not sure. I remember reading a FAQ or such that states that the natural weapon damage scales with monk unarmed damage when you have feral combat training. So taking three levels monk and the monastic legacy feat could be worth it for a synth.
Feral Combat Training: What does “with” in the Special line for this feat mean for monks making a flurry of blows?Normally a monk who has natural attacks (such as a lizardfolk monk with claw attacks) cannot use those natural attacks as part of a flurry of blows (Core Rulebook 57). Feral Combat Training allows you to use the selected natural attack as if it were a monk weapon—you can use it as one of your flurry of blows attacks, use it to deploy special attacks that require you to use a monk weapon, apply the effects of the natural weapon (such as a poisonous bite) for each flurry of blows attack, and so on.
The feat does not allow you to make your normal flurry of blows attack sequence plus one or more natural attacks with the natural weapon. In other words, if you can flurry for four attacks per round, with this feat you still only make four attacks per round... but any number of those attacks may be with the selected natural weapon.
—Sean K Reynolds, 02/15/12
Feral Combat Training and Unarmed Strike Damage: Does this allow me to use my monk unarmed damage with the selected natural attack?Yes. The feat says you can apply "effects that augment an unarmed strike," and the monk's increased unarmed damage counts as such.
—Pathfinder Design Team, 10/29/13
So, you're right that you can use your unarmed strike damage from your monk levels with your Natural Attack, but you still only get the number of attacks granted by FOB.
In order for the combination to be good you have to have enough levels in monk to get a decent number of attacks from FOB. However, the number of natural attacks a eidolon can get and the number of attacks a monk can get from FOB is both 7 at 20th level, so your maxing out at the same. With enough levels of monk you'll have a higher base damage die, but most of your damage should be coming from other things once you get to high level. And having to compromise between levels of monk of sythesist just seems like the combination wont end up as effective as it should be. The only way it seems beneficial is if you have rider effects on your natural attacks that you could flurry with that you wouldn't normally get. You could also use only 1 natural attack with your FOB (with Feral Combat training) so you could put more evolution points into things other than natural attacks, but you still have to compromise between summoner levels and monks levels to progress your evoltion points pool from summoner and your FOB and damage die from monk.
You can just trade the flurry away though. If you are playing Synthesist, you probably should. Master of Many Styles is a pretty great option.
Also, the Barbarian rage bonus is a plain morale bonus. There's nothing stopping it from applying to your modified stats as far as I can tell (unless you argue that negative effects also shouldn't apply because it's not affecting the eidolon, but the caster).
Rage would increase the summoners strength and constitution, but it specifically says you use the eidolon's physical stat scores, so it doesn't help. Your enhanced strength and con are replaced by your eidolon armor's abilities.
It replaces your base stats and then it gets buffed by a morale bonus. It doesn't permanently replace them with no way to affect them, or you'd be immune to stat lowering spells as well. The eidolon and the Summoner count as a single being for the purposes of buffs, debuffs, etc. closest FAQ I have found
Looking at it from another perspective: the eidolon only exists as part of you. If the summoner rages, so does his suit, since any effect that affects one of them, affects both (they can not be targeted individually).
I'm not sure it does. The FAQ you reference talks about spells affecting the creature, but a barbarians rage may be treated differently I'm really not sure. Synthesist summoners are a whole big bag of exceptions to the rules so I can't say for certain how it should be. In lieu of greater clarification, I still think a barbarians rage wont buff its strength.
Even if it does, your rounds of rage will be limited, and your bonuses wont increase, and you wont get many rage power unless you spend 11 levels in barbarian to get Greater Rage. The best rage powers also aren't available till level 10, and they're also duplicated by eidolon evolutions.