Does grabbing and tugging the beard playfully or calling a bearded man "father of the beard" (Abu Lihyah) constitute mockery of the beard and an act that takes one out of the fold of Islam?

The consideration in such actions lies in their motive and the intention in the heart of the doer. Such an act has become customary without disapproval. It was a custom among Arabs, when showing affection, to touch the beard of the person they were addressing. Scholars have interpreted what was narrated in the story of the Treaty of Hudaybiyyah concerning Urwah ibn Mas'ud based on this custom. Similarly, it is customary for people to identify someone by a distinguishing feature of their appearance or build, so they say: "Abu Lihyah" (father of a beard) by way of identification, just as the word "Abu" is used for someone who possesses a specific quality, like "Abu al-Karam" (father of generosity) meaning generous. Thus, using the phrase "Abu Lihyah" for a bearded person might be merely for identification and not intended as mockery.