If $f_k notto 0$ a.e., does there exist a subsequence, a set of positive measure, and $c > 0$, on which $liminf |f_{k_j}| > c$?

That's not true. For example, in $(0,1)$ take

$f_1 =1$,

$f_2=1_{(0,1/2)}$, $f_3= 1_{(1/2,1)}$

$f_4=1_{(0,1/3)}$, $f_5= 1_{(1/3,2/3)}$, $f_6= 1_{(2/3,1)}$

and so on. $f_k(x)$ does not go to 0 a.e. (the limit does not exist, for each x), but we can't find any succession that satisfies the statement, because $m(supp f_k)$ goes to zero