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.