What is the ruling on advising a non-mahram woman on matters of her studies and religion when there is no security from fitna (temptation), even with adherence to the controls of speech? And what is the precise definition of "need" for which it is permissible to speak with her? And does the fear of fitna absolutely prevent that? And what is the meaning of "no security from fitna"?

It is not permissible for a man to speak with an unrelated young woman if there is a risk of fitna (temptation/discord), whether his intention is to advise her on matters of study, religion, or anything else. The fear of fitna prevents speech, even if it is merely exchanging greetings. Al-Bukhari mentioned the permissibility of men greeting women and women greeting men when there is no risk of fitna.