Is a sampling distribution always the same as the curve for a null hypothesis?
How does the power of a z test change when sigma increases?
Assuming that they are caused solely by chance, sampling distributions will approximate a normal distribution more closely as the size of the sample increases. However, with small samples, there may be great differences.
I don't understand your second question. Sigma in a formula is used just to indicate that you add the terms that follow that symbol.
I hope this helps. If not, repost your new questions. Thanks for asking.