Is man given a choice or is he compelled in determining his destiny, whether it be to Paradise or Hell, and is a child born to a non-Muslim family predestined to enter Hell?

Man is both predestined and has free will. Allah has decreed for every newborn their lifespan, sustenance, deeds, and their ultimate destination in Paradise or Hell, as stated in the prophetic hadiths. Man acts by his own choice, and his actions are in accordance with what Allah has decreed for him out of His knowledge. Entry into Paradise or Hell is based on deeds, not lineage; thus, it is not necessarily true that the son of a disbeliever will be a disbeliever, nor the son of a Muslim will be a Muslim.