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 is His slave and Messenger.
Imaam Ahmad, Al-Bayhaqi, At-Tabaraani and others reported that Waathilah ibn Al-Asqa’ narrated that the Prophet said, “The Scriptures of Ibraheem (Abraham) were sent down during the first night of Ramadan; the Torah was sent down the sixth night of Ramadan, the Bible was sent down the thirteenth night of Ramadan, and the Furqan (i.e. the Quran) was sent down the twenty-fourth night of Ramadan.”
Some scholars criticized the chain of narrators of this hadeeth, but Shaykh Al-Albaani classified it as Hasan (good) in As-Silsilah As-Saheehah and in Saheeh wa Dha’eef Al-Jaami’ As-Sagheer.
Ibn Katheer said that the Divine Books were sent down to the Prophets in the month of Ramadhaan, as he said in his Tafseer (exegesis):
“Allah praises the month of Ramadan apart from all other months, as He had chosen it from among the months for the revelation of the Quran. And just as He singled it out with that, there is a hadeeth which states that it was the month in which the Divine Books were revealed to the Prophets.
Imaam Ahmad ibn Hanbal said, ‘Abu Sa’eed, the freed slave of Bani Haashim, reported to us from 'Imraan Abu Al-‘Awwaam, from Qataadah, from Abu Al-Maleeh, from Waathilah (ibn Al-Asqa’), that the Prophet said, ‘The Scriptures of Ibraaheem were sent down during the first night of Ramadan; the Torah was sent down the sixth night of Ramadan, the Bible was sent down the thirteenth night of Ramadan, and Allah sent down the Quran during the twenty-fourth night of Ramadan.’’
The hadeeth of Jaabir ibn ‘Abdullah stated, ‘The Psalms was sent down during the twelfth night of Ramadan, the Bible was sent down the eighteenth night of Ramadan,’ and the rest were sent down as we have already mentioned. [Ibn Mardawayh]”
Our saying that these Books were sent down in Ramadan means that the Prophets received them in Ramadan, so your second question is the same as your first question but with a different wording.
Allah knows best.