To what extent is the hadith (And even if he takes your money and whips your back) authentic, which is mentioned concerning the obligation to obey the ruler, and in which book of hadith can it be found?

This Hadith is part of Hudhayfah ibn al-Yaman's Hadith about tribulations (fitan), its origin is in the Two Sahihs with some variations in wording. It was narrated by Ibn Hibbat Allah in "Tarikh Dimashq" (The History of Damascus), and Sheikh Al-Albani compiled its narrations and authenticated this wording.

The Hadith mentions that Hudhayfah asked the Prophet, peace be upon him, about evil, fearing that he might encounter it, after people used to ask about good. He also asked him about protection from evil after good. So, the Prophet, peace be upon him, informed him about tribulations and callers to misguidance.

The Prophet, peace be upon him, advised him to adhere to the Muslim community and their leader, and to be patient with the injustice of the leader. If there is no leader or community for the Muslims, then he should withdraw from all factions, even if it means holding onto the root of a tree until death overtakes him.

The Hadith also mentions the appearance of the Dajjal after the callers to misguidance, then Jesus, son of Mary, peace be upon him, and then the establishment of the Hour.