The DF distribution and the probability of "mean reversion"

Let $y_t = y_{t-1} + e_t$, $y_0 = 0$, $e_t$ i.i.d. $N(0,\sigma^2)$, and let $\hat\rho$ be the OLS coefficient from regressing $y_t$ on $y_{t-1}$ without a constant. (a) Prove $\sum_{t=1}^Ty_{t-1}e_t = \tfrac12(y_T^2 - \sum_{t=1}^T e_t^2)$ and deduce that the sign of $\hat\rho - 1$ equals the sign of $y_T^2 - \sum e_t^2$. (b) Show that $\lim_{T\to\infty}P(\hat\rho<1) = P(\chi^2_1 < 1)$ and evaluate it to four decimals. (c) A colleague fits AR(1) models without a constant to 200 independent simulated random walks of length 1,000 and reports that "most of them mean-revert". About how many should show $\hat\rho<1$? What is wrong with the conclusion?