Is it permissible to engage in dhikr (remembrance of Allah) while watching educational lessons and performing household chores, or is complete devotion required for dhikr? And should dhikr be recited in the bathroom, as Abu Hurairah did?

Remembering Allah while performing household chores like cooking is good because it doesn't require complete devotion of the heart, combining household care with the remembrance of Allah. As for remembering Allah while watching lessons of knowledge, it is better to free one's heart for listening and learning, because learning knowledge is one of the greatest forms of remembering Allah. Indeed, everyone who works for Allah in obedience is remembering Him. It is disliked to remember Allah verbally in the restroom; it is prescribed to do so only with the heart.