I had some questions about Hadith knowledge:
1. What was the status of the people of Basrah?
2. What is the status of the narrator ‘Ali ibn Zayd? I have heard some say that he was a Shiite, so is he considered extremely weak or slightly weak?
3. Are the narrations of very weak reports written down?

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, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

The Hadeeth reporters from Basrah are like any other Hadeeth reporters; some of them are trustworthy, so their narrations are accepted, and some are weak, so their narrations are rejected. Among the trustworthy established narrators from Basrah are Musaddad ibn Musarhad, the teacher of Imaam Al-Bukhaari. Ath-Thahabi said about him, "The Haafith, the trustworthy and well-versed scholar of Hadeeth, Abu Al-Hasan Al-Asadi Al-Basri, one of the leading scholars of Hadeeth ... Those who narrated on his authority include Al-Bukhaari, Abu Daawood, Muhammad ibn Yahya, his son Yahya, Abu Zurʻah, Abu Haatim, etc."

There are many other trustworthy reporters from Basrah as well, and there are also weak, rejected ones, such as Yoosuf ibn ʻAtiyyah, known as As-Saffaar Al-Basri. Al-Bukhaari declared his narrations Munkar (objectionable). An-Nasaa'i and Ad-Doolaabi classified him as Matrook (his narrations are rejected). The same is true of Nahshal ibn Saʻeed Al-Basri, who was classified as Matrook as well.

Al-Khateeb Al-Baghdaadi narrated on the authority of Ar-Rabeeʻ ibn Sulaymaan Al-Muraadi that he said, "I heard Ash-Shaafiʻi say, '... As for the Hadeeth reporters of Basrah, they have narrated many authentic ahaadeeth with sound chains of narrations like no other narrators given their great number and widely-circulated narrations.'" [Abridged from Al-Jaamiʻ Li-Akhlaaq wa Aadaab Ar-Raawi wa As-Saamiʻ]

As for your question about ʻAli ibn Zayd, if you mean ʻAli ibn Zayd ibn Judʻaan, then he was a Hadeeth narrator from Basrah who was declared weak by Hadeeth scholars due to his bad memorization and not due to his inclination towards Shiism; he was slightly inclined towards Shiism.

Ath-Thahabi wrote about him:

"He is ʻAli ibn Zayd ibn Judʻaan At-Taymi, the Imaam and knowledgeable scholar, Abu Al-Hasan Al-Qurashi At-Taymi Al-Basri ... He narrated reports on the authority of Anas ibn Maalik, Saʻeed ibn Al-Musayyib, Abu ʻUthmaan An-Nahdi, and ʻUrwah ibn Az-Zubayr... Those who narrated on his authority are: Shuʻbah, Sufyaan, and Hammaad ibn Salamah... He was a well-versed scholar, although he was slightly inclined to Shiism and his memorization was not precise. Abu Zurʻah and Abu Haatim said that he was not a strong reporter. Al-Bukhaari and others said that his narrations are not to be used as evidence. Ibn Khuzaymah said, 'I do not use his narrations as evidence due to his bad memorization.'" [Abridged: Siyar Aʻlaam An-Nubalaa']

As for the very weak reports, if you mean the ahaadeeth that are declared very weak by Hadeeth scholars, then know that they are written down and documented along with the clarification of their weakness.

Allah knows best.