logically, given A asked B if he would mind helping the answer 'sure' would seem to suggest that B does in fact mind. However, that is not how this response would be interpreted by a native speaker. ' Would you mind helping' is a polite way of asking 'will you help me' and so the 'sure' is really in response to the meaning behind the question, rather than a literal response to the wording. Sure is a very casual response. A more formal (and literal) response to the same question might be 'not at all' - as in, no, I don't mind helping at all.

"Sure" is simply an expression of acceptance. Colloquially, it's used quite a lot and less often in formal situations. Normally, you would expect the answer to be "no" in response to the question "do you mind," because "to mind something," means "to worry about" or "to be annoyed with." So I understand your confusion in this case. Despite the meaning of "to mind," colloquially you will hear the word "sure" from time to time. If you are uncertain of the speaker's intentions through their words alone, pay attention to their tone of voice and facial expression to confirm what you think they mean. If, for some reason, you are still unsure, then ask directly to confirm. I hope this helped. Let me know if you have any more questions.

In this context, when they say "mind" it really means: are you able to do me a favor, does it bother you to do me a favor, or can you do me a favor? The equivalent of this in Japanese would probably be, dekimasuka? however I'm not a fluent Japanese speaker but this is what I can assume.