Is the Duha prayer an obligatory Sunnah or a Fard (compulsory act), and what are its times and conditions, and is it sufficient in place of the Fajr prayer if it is missed?

The Duha prayer is a confirmed Sunnah (supererogatory act), not an obligatory one. This is evidenced by the Prophet's (peace and blessings be upon him) conversation with the Bedouin when he asked him about Islam. The Prophet (peace and blessings be upon him) enjoined and encouraged it. Its time and conditions are the well-known conditions for prayer. The Duha prayer does not compensate for the Fajr prayer; for the obligatory (fard) is the root and the supererogatory (nafl) is a branch, and the requirement of the obligatory prayer is not waived by performing the Duha prayer.