I guess I could do it only once, as I have to spend a swift action.
This is the case. When one of your attacks hit, you spend the swift action to activate your other ability. Your other hits technically qualify, but you don't have more swift actions to spend, so you can't activate it more than once in a round.
Yep Betasprite has it. The good news is that since you wait till you hit to decide to use it you won't waste your swift if you miss an attack.
No in the case of you growing claws you wouldn't get a free attack the effect of that power is to grow claws. At best that bloodline might let you grow your claws as a swift after an attack. The feat is for use with targeted powers such as
Electricity Ray (Sp): Starting at 1st level, you can unleash an elemental ray as a standard action, targeting any foe within 30 feet as a ranged touch attack. This ray deals 1d6 points of electricity damage + 1 for every two sorcerer levels you possess. You can use this ability a number of times per day equal to 3 + your Charisma modifier.
Well, the bloodline power just gives you the natural attack: you still have to hit with the claws to do damage, so I would agree with your GM here.
I think Feral Combat training is more likely to get you what you want, especially if your GM will let you flurry with your claws once you grow them.
Yes with feral you could punch if you hit grow claws then use your claws on remainder of the flurry.
a better option to consider would be to NOT flurry. If you simply use a single unarmed strike, hit, and then grow claws, you can at that point make your claw attacks (which would be 3 total attacks). The reason this works is because during a full-attack action you can use natural attacks that aren't required for other attacks, and since unarmed strike allows you to use body parts other then your hands to attack, you can do this.