Is it obligatory to pass the hand over the eyes, the chin (al-anfaqah), between the nostrils and their sides, and the lips during ablution, or does the water reach these areas without that? And is it obligatory to place a handful of water under the chin to ensure water reaches the entire face?

You are not required to pass your hand (rub) over your eyes or other washed parts during ablution (wudu') and ritual bathing (ghusl), according to the majority of scholars, if you are certain that the water has reached [the area], or if you have a strong presumption that it has. The obligation is to ensure that water reaches the entire face. However, if water cannot reach a part except by passing the hand over it, then it becomes obligatory.

It is not obligatory to take a handful of water and place it under a thin beard. Rather, what is obligatory is to comb through it (takhleel). This is achieved by pouring water over the face and passing the hand while combing through, which means ensuring water reaches the skin if the beard is thin enough for the skin to be visible.