Sure is used to signal consent, or to put it another way, willingness to go along with something. It's often used in response to requests for permission:
Alice: Would you mind if I take the car?
Bob: Sure, go ahead.
In the above, Bob is willing to go along with the proposition of letting Alice use his car. Bob is not signalling literal agreement with Alice's words, which is exactly what yes does:
Alice: Would you mind if I take the car?
Bob: Yes, I would mind very much. Take the bus.
Compare the following:
Alice: Would you mind if I take the car?
Bob: No, go right ahead.
In this case, literal disagreement has the same illocutionary force as sure, so we can see that yes and sure don't mean the same thing. However, the opposite might be true in another situation:
Alice: Would you like some cake?
Bob: Yes, I'd love some.
Alice: Would you like some cake?
Bob: Sure, thank you.
Here, Bob's willingness to go along with Alice's proposition has the same force as literal agreement. No would signal the opposite:
Alice: Would you like some cake?
Bob: No, thank you.
In other words, sure signals consent, while yes and no signal literal agreement or disagreement.
The easiest way to understand this usage of Goodwife is to think of it as "Mrs." but for married women (commoners) in the old days. According to Wikipedia,
Goodwife (Scots: Guidwife), usually abbreviated Goody, was a polite form of address for women, formerly used where "Mrs.", "Miss" and "Ms." would be used today. Its male counterpart is Goodman.
Now, let's focus on the part you do not understand: a married woman not of noble birth.
You can understand this phrase as a married woman (who is) not of noble birth. But what does not of noble birth mean? According to Macmillan Dictionary,
noble (adjective)
2. belonging to the highest social class. In the U.K., noble people usually have a title, for example Duke or Baroness
of noble birth/descent/blood: a young man of noble birth
So, not of noble birth means "not belonging to the highest social class". Thus, a married woman (who is) not of noble birth would mean "a married woman who is not belonging to the highest social class".
Bonus: How to understand the definition
2. Goodwife Used formerly as a courtesy title before the surname of a married woman not of noble birth.
= "Goodwife" (the word) was used "formerly" (in the past) as "a courtesy title" (a polite title, such as "Mr.", "Mrs.", etc.) before the "surname" (the last name or family name) of a "married woman" (a woman who is not single) (who is) "not of noble birth" (see the definition of "of noble birth" in my explanation above).
To make the parsing a bit clearer, here is how you can read it:
Goodwife: [ Used formerly [ as [ a courtesy title ] before [ the surname of [ a married woman [ not [ of noble birth ] ] ] ] ] ].
Best Answer
It does indeed mean no, and this isn't even just "Hagrid-speech"; nah is a common informal way to say "no".
I'm not sure where you mean it should have been placed in the previous sentence, but I'll attempt to explain why it appears where it does.
There's an implication there in the nah that's based on prior context."The gold isn't in the house; there's no reason to go to the house. No, that's not where we need to go; the first stop for us is Gringotts!"
Basically in this case, the nah is being used to say "No, not that thing you just said. We're doing this." Another example: