Convergence in Norm to Zero If a sequence, $\{f_n\}$, of nonnegative functions with $\lim_{n\to\infty}\int_{X}\sqrt{f_n}dm=0$, must it be the case that $\lim_{n\to\infty}f_n=0$ $m$-a.e. on $X$?
No, though  $\liminf_{n\to\infty}f_n=0$, $m$-a.e., by Fatou's Lemma; and also, $f_n\to 0$ in measure, by Markov's inequality. For a counterexample, use the "typewriter sequence" of functions (Example 4 in https://terrytao.wordpress.com/2010/10/02/245a-notes-4-modes-of-convergence/)
Yes. The definition of the $L^2$ norm is $\|f\| = (\int_X f^2\,dm)^{1/2}$, so your hypothesis says that $\|f_n^{1/4}\| \to 0$. Since pointwise multiplication of functions is continuous in the 2-norm, it follows that $\|f_n\| \to 0$ as well. Since the $0$ element of $L^2$ is the class of functions that are $0$ $m$-almost everywhere, this gives you your answer.