Is it permissible to eat the food distributed during the Mawlid (birthday) of the Prophet, peace and blessings be upon him, and is it true that Abu Lahab's torment was lightened every Monday because he freed the slave girl out of joy for the Prophet's birth?

There is no such thing as "Mawlid al-Nabi" (the Prophet's birthday) in Islam. It was unknown to the Companions, the Successors, and the Imams; rather, it was innovated by some innovators. All actions specific to this day, such as holding celebrations and distributing food, are considered forbidden acts, as they are a revival of an innovated holiday. Whoever participates in this by eating the food or sitting at the tables is complicit in reviving the innovation and assisting in its establishment, due to the Almighty's saying: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression) Al-Ma'idah/2. Scholars have issued fatwas prohibiting eating the food distributed during such innovated holidays, even if the slaughtering was for consumption. One should not eat from it and should not attend these celebrations as a form of disapproval, unless the attendance is for advising them without participating.