Is the ḥadīth that mentions that whoever listens to a lesson of knowledge in the mosque or leaves [his home] in a state of ritual purity for the obligatory prayer will have the reward of a Ḥajj, authentic?

The reward for leaving one's home in a state of purity to perform an obligatory prayer is equivalent to the reward of a pilgrim in the state of ihram. As for one who prays Fajr in congregation, then sits remembering Allah until the sun rises, and then prays two rak'ahs, he will receive the reward of a complete Hajj and 'Umrah. It is hoped that whoever prays Fajr in the mosque and sits in a study circle will attain this reward, as it is considered a circle of remembrance of Allah.