You are right. If you ask a question with 'would/do you mind', the positive response has to be in a grammatically negative form: Mind some advice? Of course not. (meaning 'Of course I don't mind' = 'I'm happy to hear some advice.') To respond with 'Sure', you'd have to swap the question round, and ask something that requires a grammatically positive response. For example: Can I give you some advice? Sure.

You are technically right but she is colloquially right. SuKi is right above, but many people who speak English natively in the US will answer with "sure".

Sure = of course in this situation E.g. Can I have another glass of your fine whisky? Sure / Of course. "Of course not" has the opposite meaning.

Sorry for my carelessness.

If answering a yes/no question, 'Sure' means yes; 'Of course not' means no.