I think the hyperbolic tangent is the function you are looking for:

tanh(x) = (1-exp(-2x)) / (1+exp(-2x))

Whether it will affect my networks ability or not?

In general it is hard to say whether using tanh with outputs in the range [-1 1] or another sigmoid with outputs in the range [0 1] would affect performance. I don´t think it will make much of a difference, but I guess to try both of them out is the best.

Sigmoidal fuction is correct because you behavior derived all behaviors

Just to add on the previous comments. Any smooth function can be used as an activation function, especially in the hidden layer. My suggestion is that if you want your output to be in the range [-1,1] then you could use a hyperbolic tangent function, while if you want it to be in [0,1] you could use a logistic sigmoid. This avoids the need for post-processing. If, however, the output range is something else then whichever function you use, you need to do some post-processing.

I am not sure I understand what you mean about error function, but an error function (or cost function) is a measure of how well your network has learned the data. If your prediction problem is a regression one (i.e. predicting a continuous value, e.g., the price of houses) then the Mean Squared Error is a sensible cost function. If your problem is classification (i.e. predicting whether your input belongs in a certain class, e.g., whether a patient has cancer or not), then the Cross Entropy error function is more appropriate. In the latter case you should consider using a softmax activation function, as it makes your outputs get interpreted as probabilities (each output is between 0 and 1 and the sum of all outputs is 1).

If you want to use an optimizer that needs gradient information, such as the standard backpropagation algorithm (which basically calculates the first derivatives and does gradient descent), then the derivatives will depend both on the error function and the activation function of the corresponding neurons.

I am not sure whether I covered your question. Let us know if something is unclear.

RMS Root Mean Square for computing the error.

Make use of tan-sigmoidal activation function which has range between -1 to +1. Else normalize your input data between 0 to 1 & use log-sigmoidal function.

its always a hit and trial method to choose the best for a particular application. You will find in some cases the either one performing good for similar situations and normalization ranges.

It depends on problems.