Is true that the sampling distribution of $\ln \left(\chi^{2}\right)$ converges to normality much faster than the sampling distribution of $\chi^{2}$? 
If true is the consequence true that $X \sim \chi^{2}(k)$ then $\sqrt{2 X}$ is approximately normally distributed with mean $\sqrt{2 k-1}$ and unit variance?


Also true that If $X \sim \chi^{2}(k)$ then $\sqrt[3]{X / k}$ is approximately normally distributed with mean $1-\frac{2}{9 k}$ and variance $\frac{2}{9 k} \cdot[16]$ This is known as the Wilson-Hilferty transformation, see (18.24), p. 426 of Johnson. ${ }^{[4]}$
This normalizing transformation leads directly to the commonly used median approximation $k\left(1-\frac{2}{9 k}\right)^{3}$ by back-transforming from the mean, which is also the median, of the normal distribution?

I am wondering if this is true.
If this is true, then I heard the root of a chi-square converges faster to normality.
https://en.wikipedia.org/wiki/Chi-squared_distribution

I will only answer part of question and only as a hint:

then I heard the root of a chi-square converges faster to normality.

Since the chisquared distribution is right skewed (less so for high df) this sounds reasonable, and below is how you can investigate it yourself. Below a plot of densities of a chisquared distribution, and the density of log of random variable with same chisq dist: (you do the same with square root, and other df)

From comparing the density functions you can answer yourself!
Below the R code used:
make_dens_of_log <- function(df , ...) {
  function(t) exp(t) * df(exp(t), ...)
}

plot( make_dens_of_log(dchisq,10), from=0, to= 20)
plot( function(x) dchisq(x,10), from=0,to=20,col="red",add=T)
title("density of chisq(red) and log of chisq\n   (10 df)")

