Is it permissible for scholars to criticize one another, and do they have a precedent or chain of transmission for this that reaches the Prophet, peace and blessings be upon him, and must the words of any of them be affirmed?

The mutual advice and discussion among scholars are praiseworthy in Islamic law. As for one scholar demeaning his brother, it is generally forbidden, unless there is fear that his mistakes might be emulated, in which case the legal ruling should be clarified without disparagement. If a scholar disparages others, then students of knowledge should not spread that, for the words of peers about each other are to be folded away and not narrated.