Aperson who gives you a counterargument is disagreeing with you.

Why would you include a counter-argument in your essay? Doesn’t that weaken your argument?

Actually, no. Done well, it makes the argument stronger. This is because it gives you the chance to respond to your reader’s objections before they have finished reading. It also shows that you are a reasonable person who has considered both sides of the debate. Both of these make an essay more persuasive.

the narrator is saying that the old lady made her feel uncomfortable. you know this because they say "i found that rather irksome", "perhaps it was her being so upright and trim", and "i don't think it was that, because i thought that quaintly pleasant." hope this !