convergence in distribution and convergence of quantile
Solution 1:
Yes. If $X$ is a random variable with distribution function $F$, then for $0<p<1$ define the quantile function as $Q(p)=\inf(x: F(x)\geq p)$. Then $X_n\to X$ in distribution if and only if $Q_n(p)\to Q(p)$ at all continuity points $p$ of $Q$.
Added: It's a nice exercise to prove this result from the definition. On the other hand, it is Proposition 5 (page 250) in A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray, and is also proved in Chapter 21 of Asymptotic Statistics by A. W. van der Vaart.