What is the ruling on distributing pastes and sweets on the first Sunday of Sha’ban, and distributing Hareesah on the day of Ashura, with the intention of imitating the Prophet Zechariah or to achieve one's desires and vows?

What you mentioned about the actions on the occasion of Zakariya and the distribution of hareesah on the day of Ashura are innovations (bid'ah) that are not found in the Sharia, nor were they known to the righteous predecessors. These are innovations that the Prophet, peace and blessings be upon him, warned against if those who practice them intend to draw closer to Allah Almighty through them. It is incumbent upon a Muslim to avoid them and to keep others away from them as much as possible.