Volatility of volatility and the square root

Under the pricing measure, a stock's instantaneous variance $v_t$ follows a driftless geometric Brownian motion,

so volatility starts at 20%. The volatility of variance is $\xi = 0.8$, with time in years.

(a) Use Itô's lemma to find the SDE for the volatility $\sigma_t = \sqrt{v_t}$. What are its drift and its volatility?

(b) Find $E[\sigma_1]$ and compare it with $\sqrt{E[v_1]}$.

(c) Find the median of $\sigma_1$. A trader argues that variance is a fair game here, so volatility must be one too. What is wrong with the argument?