Why is eulers number used as a constant in sigmoid
Euler's number pops up in a lot of places naturally; not quite something to do with growth rates but arises easily in common limits. 

The form of the sigmoid function wasn't chosen because its derivative has a nice property, although that's true. It also wasn't chosen because it's a function with range (0,1); many functions do that.

The sigmoid function arises because it's the correct answer to a common type of problem. We think of logistic regression as a common classifier predicting a class probability, but it really is regression under the hood. It's not regressing the probability though, but log-odds of the probability (logit function). This is the right way to apply regression, given assumptions about the distribution of errors in a classification problem, which aren't the same as in a simple linear regression.

The sigmoid function is the inverse of the logit link function. That's why it's there. It gets from the regression output to the actual desired output, a probability. The logit function is there because it is implied by the assumption about the distribution of the 0/1 dependent variable.

That's actually it. It has to be there in certain types of common problems because it's the answer to those problems, not because it was chosen for nice properties.