Ok, not "trial" is the unit of observation, but "child" is (n=30). One could compare the actual number of hits for each child with expected numer of hits using one-sample Wilcoxon signed rank test.Q1a. whether the children make more hits under experimental condition than chance level,

n=30 chlidren, each child with 2 measurements (number of hits under experimental condition and under control condition, respectively). Wilcoxon signed rank test could be an option.Q2. for each child, whether (s)he is more likely to "hit" under experimental condition than under control,

Yes. They turned 30 subjects into 300 subjects, which isn't correct.But the data do not come from 300 subjects, and instead, each subject was observed multiple times, within and across cells. I think underlying assumptions for binomial test are violated.

With kind regards

Karabiner