Is it obligatory to rub a thick beard with gaps on the sides when performing ablution (wudu) and ritual purification (ghusl), or is it sufficient to wash it along with the face once, and similarly for facial hair (eyebrows, mustache, and the hair below the lower lip)?

Rubbing (Idlak) is desirable in ablution (Wudu') and ritual bathing (Ghusl) according to the majority of scholars, contrary to the Malikis.

As for facial hair, if it is thin and the skin beneath it is visible, it is obligatory to make the water reach both its outer and inner parts. If it is thick and the skin beneath it is not visible, it is obligatory to wash only its outer part, and it is desirable to run water through it (Takhleel) in ablution. However, in ritual bathing (Ghusl), it is obligatory to wash both its outer and inner parts.