Update (April 21, 2017): answer to question #3 The convergence rate of uniform order statistics to their expecations obtained from $(1)$ is NOT optimal for order statisic $X_{(k)}$ for which $k=o(n)$ or $k/n \to 1$. However, it is optimal for $X_{(k)}$ for which $k= c_n n$ with $$1 > \limsup c_n \ge \liminf c_n >0.$$ In other words, close to the edges 0 or 1, the optimal reate of convergence is proportional to $$\frac{\log n}{n},$$ whereas in other parts of the compact interval $[0,1]$ the optimal rate of convergence is $$\frac{\sqrt{2 \log_{(2)}n}}{\sqrt{n}}.$$ This can be seen from:

Update (April 21, 2017): answer to question #3 The convergence rate of uniform order statistics to their expecations obtained from $(1)$ is NOT optimal for order statisic $X_{(k)}$ for which $k$ or $n-k$ is bounded. However, it is optimal for $X_{(k)}$ for which $k= c_n n$ with $$1 > \limsup c_n \ge \liminf c_n >0.$$ In other words, close to the edges 0 or 1, the optimal reate of convergence is proportional to $$\frac{\log n}{n},$$ whereas in other parts of the compact interval $[0,1]$ the optimal rate of convergence is $$\frac{\sqrt{2 \log_{(2)}n}}{\sqrt{n}}.$$ This can be seen from:

Update to question #3 (April 21, 2016)

Update (April 21, 2017): answer to question #3 The convergence rate of uniform order statistics to their expecations obtained from $(1)$ is NOT optimal for order statisic $X_{(k)}$ for which $k=o(n)$ or $k/n \to 1$. However, it is optimal for $X_{(k)}$ for which $k= c_n n$ with $$1 > \limsup c_n \ge \liminf c_n >0.$$ In other words, close to the edges 0 or 1, the optimal reate of convergence is proportional to $$\frac{\log n}{n},$$ whereas in other parts of the compact interval $[0,1]$ the optimal rate of convergence is $$\frac{\sqrt{2 \log_{(2)}n}}{\sqrt{n}}.$$ This can be seen from:

Lemma A2.1 on page 148 of "asymptotic expansions for the power of distribution free tests in the one-sample problem" by Albers, Bickel and Zwet.

Update to question #3 (April 21, 2016)