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.
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.
They didn' keep their gold in the house, boy! Nah, (the) first stop fer us is Gringotts.
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:
Amy: "Did you try that new Italian place on your date last night?"
Mary: "What, and completely ruin my diet? Nah, Mark took me to a sushi place instead."
Best Answer
It is a recent coinage, with very little currency. I.e almost nobody will know the word, and you won't be understood if you use it. The earliest use I can find on the internet dates from 2005, by Chris Lehmann in a review of Sam Harris's The End of Faith
It is a portmanteau of Believe and Bourgeoisie. The Bourgeoisie are, in Marxist use, the middle classes who own much of the wealth (contrasted with the proletariat or working classes, and the aristocracy). The Believoisie are the people who believe in God, and are said to control the morals of society in the way that the Bourgeoisie control the Captial. As a portmanteau, the pronunciation would be something like /bɪˈliːvwɑːˌziː/, that is "believe" followed by the last two syllables of "bourgeoisie".
In the American context, it refers to the "religious right" in America, that have influence over many politicians, particularly Southern Republicans
The implications are negative, just as Marxists use "Bourgeoise" as a term of abuse.