What is the ruling on someone who forgot a Sunnah in ablution, and is their ablution valid? How do they rectify it in the following cases: if they remember during ablution, or after finishing but before the limbs dry, or after they have dried? And what if what was forgotten was an obligatory act (Fard)?

Forgetting a sunnah (definition) (recommended act) of Wudu' does not invalidate it or affect its correctness. If one remembers it during Wudu', they should not interrupt their Wudu'; rather, they should complete what they are doing and then perform it. The relied-upon view is that they should perform it after finishing Wudu'. If one remembers it after completing Wudu', they should perform it for the prayers they will perform, whether they remember it before their limbs dry or after. However, some sunan are not performed if forgotten, such as washing the hands up to the elbows, repeating the wiping of the head, blowing the nose (istinthar), and renewing the water for wiping the ears, because performing them again would lead to a disliked act (makruh (definition)).