Is the manner in which I recite the morning and evening adhkar in the mosque after the Fajr and Maghrib prayers correct, and is it better to recite them at home or in the mosque, and is it permissible to sleep after the morning adhkar?

It is better to offer the post-prayer adhkar first, as they are restricted to immediately after the prayer. Then, one can recite the morning or evening adhkar. It is permissible to recite them in the mosque or at home, though reciting them in the mosque is better, especially the morning adhkar, due to the hadith of Anas ibn Malik: "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of a complete Hajj and 'Umrah." He (the narrator) said: The Messenger of Allah, peace and blessings be upon him, said: "Complete, complete, complete." There is no harm in sleeping after reciting the morning adhkar, but it is better to occupy oneself with general dhikr or Qur'an recitation until sunrise, then pray two rak'ahs.