I am not completely sure, but I would say that the apothecary suggests that Jane go to school because he believes Jane would be happier and healthier away from Gateshead. She tells him how unhappy she is, so that is why he wants to help her.
The adverb would probably be secretly
I find all of the answers pretty neutral (that is, giving only facts, no judgements) except on sentence 3: this is because of the word "sharply".
It seems that the author of this sentence makes a judgement about the split: that they're very split, that their argument was very intense. It seems like a judgement to me more than the other sentences.
