What was the reason for the conflict between Ibn Az-Zubayr and Marwan ibn Al-Hakam or Yazid? Is that considered a rebellion against the ولي الأمر (ruler), and is Hussein considered to have rebelled against Yazid?

Summary of the answer regarding the conflict between Abdullah ibn al-Zubayr and Yazid ibn Mu'awiya:

A conflict arose between Abdullah ibn al-Zubayr and Yazid ibn Mu'awiya due to Ibn al-Zubayr's refusal to pledge allegiance to Yazid as Caliph. After Yazid's death, people pledged allegiance to Ibn al-Zubayr as Caliph. Then Marwan ibn al-Hakam rebelled against him, followed by his son Abd al-Malik, until the Caliphate was restored to the Umayyad household. Some researchers and historians consider Marwan ibn al-Hakam and his son Abd al-Malik as rebels who revolted against the Commander of the Faithful Abdullah ibn al-Zubayr, and that Ibn al-Zubayr was the Imam after the death of Mu'awiya ibn Yazid. Ibn al-Zubayr's goal in his opposition was to restore the Ummah to a life of consultation (shura) and to ensure that the best among them governed the affairs of the Ummah. There is a difference of opinion regarding the permissibility of Al-Husayn's and Ibn al-Zubayr's revolt against Yazid: Some permitted revolt against an unjust Imam, and they believe that Al-Husayn revolted to establish justice. Others believe that the texts from Imam Ahmad indicate that it is not permissible to revolt against an unjust Imam. Ibn al-Jawzi said: The statement that Yazid was in the right and Al-Husayn was mistaken is a commoner's opinion, for Yazid's actions of plundering Medina, bombarding the Kaaba, and killing Al-Husayn necessitate the annulment of his allegiance. Al-Ruhaibani said: Al-Husayn's revolt against Yazid was permissible, and Al-Husayn was in the right based on his ijtihad (independent reasoning). Ibn Hazm held that Yazid ibn Mu'awiya, Marwan ibn al-Hakam, and Abd al-Malik ibn Marwan had no valid interpretation for their revolt against Ibn al-Zubayr; rather, it was pure rebellion. As for Ibn Hajar al-Haytami, he believes that the prohibition of revolting against an unjust Imam came after the era of the Companions and the stabilization of affairs, thus it does not include the revolts of Al-Husayn and Ibn al-Zubayr. Ibn Abbas and other Companions disapproved of Al-Husayn's revolt against Yazid out of fear for him, but the Ahl al-Sunnah (Sunnis) despise Yazid.