Assalaamu alaykum. While I was reading the translation of the book Al-Adab Al-Mufrad, I came across the hadith in which ‘Aa’ishah bint Talhah said, "I spoke to ‘Aa’ishah when I was under her protection. People used to visit her from all places. Shaykhs used visit me because of my place with her. Young men used to treat me as a sister and give me presents and write to me from their cities. I said to ‘Aa’ishah, 'Aunt, this is a letter from so-and-so and his gift.' ‘Aa’ishah told me, 'My girl, answer him and reward him. If you do not have anything to give, I will give you something.' She told her to give it to her."
I am confused that:
1) What does she mean by visiting her. I do not know the age of the narrator, but what is the islamic position regarding men visiting non-Mahram (permanently unmarriageable) women? I heard that interaction with non-Mahram women without any reason is prohibited. So why did the Shaykhs visit her? It would be great if you could provide some guidlines for interactions with the opposite gender, which is unavoidable in many Muslim and Non-Muslim majority countries. And even if it is avoidable, like in the times of the Companions, what makes it permissible?
2) The narrator said that the young men used to write letters to her, also, the chapter's name by Imam Al-Bukhaari, where he used the word women, suggests so. What makes such interactions permissible? I mean, is the devil not the third when a young man and a young woman are secluded? What are the implications of this fact with regard to modern-day chatting on Facebook or WhatsApp? Please let me know.

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

This report was cited by Al-Bukhaari in his book Al-Adab Al-Mufrad, as you mentioned, and Shaykh Al-Albaani classified its chain of narration as Hasan (good) in his book Saheeh Al-Adab Al-Mufrad.

Regarding the first question, the report is not problematic, whether with regards to the visits or the gifts. Both actions (visiting and offering gifts) are permissible provided that there is no fear of Fitnah (temptation) and in the absence of suspicions. This is evidenced by the story of Abu Bakr and ʻUmar visiting Umm Ayman after the death of the Prophet, sallallaahu ‘alayhi wa sallam, which we mentioned in fatwas 36027 and 314452.

The permissibility of offering gifts is deduced from the hadeeth narrated on the authority of Anas ibn Maalik who related that a Jewish woman gifted the Prophet, sallallaahu ‘alayhi wa sallam, a poisoned sheep and he ate from it. Also, it is further evidenced by the general indication of the hadeeth cited by Al-Bukhaari: "Exchange gifts so that you would love one another." [Al-Adab Al-Mufrad]

As for the second question, the report does not suggest anything related to the prohibited mixing between men and women. As for writing or talking between a man and a non-Mahram (marriageable) woman, they are also permissible as long as the Shariah guidelines are observed. Al-Bukhaari's remark in his commentary on the report indicates that he held that such interaction is permissible.

If there is no need for interaction or if the Shariah guidelines are not observed, it is impermissible to write or talk to each other, as this may lead to Fitnah. However, it does not necessarily entail being in seclusion with a non-Mahram woman.

Allah knows best.