Standard deviation after a non-linear transformation of a normal distribution

Question

I have two questions for you that have been bothering me for quite some time now.

For the first one

I have a normal distribution (density function f(x)) on which I only now the mean and standard deviation.

I have to apply a non-linear transformation over the variable x, let's call k the new transformed variable, defined as: k = x ^ -2

If x_mean is the mean of my first normal distribution, then can the new mean be calculated as : k_mean = x_mean ^ -2

Now, for the standard deviation I guess it's a bit different, here's what I did:

k_std = k_mean * (x_std / x_mean)

Does it make sense to you? Let's suppose (x_mean / x_std)*100 = 7%, then I assumed that (k_mean / k_std)*100 = 7%. What do you think?

For the second question

I have a square of length 2a in which they are random points. I need to know the mean and std of the induced normal distribution of these points. However, I only know the mean and std of small parts of this square.

example

I know how to sum up all the small square mean to get the total one. However, for the std I have some doubts.

Here's what I would do (btw, in this example all the small squares have an equal surface, but it may not be always true):

std equation

What do you think?

Thank you for your help!