What is to be done concerning the contradiction between Sheikh Al-Albani's authentication of hadiths and others' deeming them weak? Is his authentication always binding, or is it permissible to deem weak what he has authenticated?

Correcting and weakening hadiths are matters of ijtihad. A Muslim, when faced with differing scholarly opinions on such matters, should follow a trusted scholar if they are not proficient in the science of hadith themselves. Sheikh Al-Albani is among those deeply rooted in the science of hadith, but he is not infallible. Sheikh Al-Albani graded the hadith of Asma as حسن (good), whereas we have explained its weakness in previous fatwas.