Alright, here we go:
How Cyberlimbs affect your Attributes:
Cyberlimbs do NOT affect your Physical Limit. If your 'whole' strength is 2, your reaction is 2, and your body is 2, then your Physical Limit is 3, even if you have Strength 9 on every arm and leg and cyber torso.
When using a skill that involves only the limb in question, you use the Attribute of that limb. Shooting an SMG with your right cyberarm allows you to make use of the agility of that cyberarm. Punching with your cyberarm gives you damage based on the strength of the cyberarm.
When using a skill that makes use of multiple limbs, you use the average attribute of the limbs in question. Agility 2 with a right cyberarm agility 8 shooting a sniper rifle acts as if he has Agility 5 for the test.
When using a skill that requires the 'careful coordination' of many limbs, which I read to mean you put most of your body into it... leaping across gaps, doing somersaults, etc... you use the lowest Attribute in the equation (probably your natural, but you may have bought a weak cyberarm or something).
That's for full limbs. Partial limbs, hands, and feet only apply that attribute to skills that reasonably only use that part of the body. A cyberhand gives you no bonus to firing an smh or rifle (though some may allow it to apply to a pistol). However, to pick a lock with that hand, you would get the hand's agility.
Clear as mud?