Learn English – What does “A man passed him, and a man and a woman, talking” mean

maybe he'd wake Dudley up, just to annoy him.

See it is an idea that came to his mind as a wish, but he intended not to do that. "Would" is used here to express that he will do something in future (it happened in past of now). So,

... he'd wake Dudley up, just to annoy him.

it means "He will surely wake Dudly up" (in past of now). But he did not actually intend to do so. It was just a quick thought. So "maybe" is used to make the sense a "wish" or "possibility". So when the sentence is rewritten as:

maybe he'd wake Dudley up, just to annoy him.

it consists of the whole sense i.e. he is not surely going to Dudly up, but he is just having a quick thought of it.

Thanks for supplying the context. You're correct in your understanding of "over his milk".

The oxcart question is not offensive in itself, though it is colloquial and hence sounds a bit silly.

The key sentence in the larger context you provide is "She had, for almost two years now, baited him. In the halls, in the elevator, even in his own office, into which she romped now and then like a circus horse, she was constantly shouting these silly questions at him."

To bait someone is to deliberately taunt or annoy them, hoping to get a response from them. (Analogous to offering bait to a fish in the hope that it would bite.) She behaves extravagantly ("like a circus horse"), she shouts out rhetorical questions that are both colloquial and hackneyed, and ultimately she is always asking the same question that assumes things are going badly. She's negative, imposing, and annoying.

At least that's my take on it, based on a few paragraphs. I have to admit that I've never read that classic story before.