After that techno thread on a very similar topic this set of question was basically unavoidable.
Ok so before we get started lets cover a couple things by definitions
Cyber-limbs rules recognize 3 categories relating to this, and each one has to be addressed.
1. Full Limb- The whole arm, from tips of the fingers to the shoulder. This one is well covered, we have good examples of how it should work, and its fairly easy to understand
2 Lower Arm- From tips of the finger to the elbow. This one is not well covered, it counts as half a limb for health box purposes, we know how much capacity it has, but when it can be used on its own is remains fairly theoretical.
3 Hand- From tips of the fingers to the wrist. This one hasn't really been discussed at all. We know stats from a table and that's really about it.
So the question is when can you get away with using just the limbs stats, (Str, Agi), and when you must average it. (the average being the average of the limbs stat and the character's normal stat)
For the full arm if you are employing one handed weapons, you use just the limbs stats (melee ranged etc). The book is clear on that.
For the Half limb, in the last thread I purposed when using only pistol sized weapons, specifically excluding melee you could use just the limb. The logic being you are getting about half the effect of the full limb. Exactly what constitutes a pistol sized weapon still needs a little more specification but the intent is clear, and is my attempted to be consistent with the example given.
For just a Hand I would say if your using a weapon your going to have to average it with the character's normal stat. This really isn't a bad thing, if your cyberhand is agi 9 and your regular stat is agi 3, your going to be shooting at agi 6. So in effect your getting a 3 die bonus from .25 worth of essence. But that is
RAW as I understand it.
Now to specific questions when you say Cyberhand I am assuming just the hand, if you mean lower arm most of these answers are mostly the same, only a couple things are different.
1. A cyberhand is stated in the core and it has different capacity, essence cost, and availability, you can see them on page 457 of the core, there are obvious common sense limits on the differences. For example you can't put a cyber gun slide in a hand.
2. A hand's capacity is high enough to take a full rating 3 enhancement, but only in one stat. So agi 9 is possible or str 9 is possible but not both.
3. As explain above it is not the same in all ways, it has some similarities but is subject many more limitations, the full limb's much great capacity means achieving full maximum is possible, that is not the case with a single hand.
4. For a two handed melee attacks your going to be running the average, even if you had both hands replaced.
5. For a two handed gun your also going to be running the average, even if you had both hands replaced.
As I see it if your making a cyber-primary street sam and your intent on doing the cyberlimb method, your best bet is to get a full limbs, it gives you the most options, and lets you get the maximum stat advantage. If your looking to avoid essence loss and are ok with being a little less effective from the combat stand point, a half limb, maybe a good option for you. If your looking to mess around with builds are trying maximum effect for minimum essence cost a hand is an interest point of discussion, but its going to lag behind a full limb in terms of attribute advantage and total options.
Of course mileage and opinions will very, and your GM will be the final arbiter.