Is Duha Prayer obligatory every day, and is it true that the Prophet, may Allah bless him and grant him peace, used to pray it some days and leave it some days?

Duha prayer is a confirmed Sunnah, not obligatory. Two rak'ahs are sufficient for it, and whoever prays it attains great goodness and reward, while whoever leaves it incurs no sin. The Prophet, peace and blessings be upon him, enjoined it upon Abu Hurairah, may Allah be pleased with him. The Prophet's inconsistency in performing it was out of fear that it might become obligatory upon the Ummah. Aisha, may Allah be pleased with her, was constant in performing it.