Short answers, pulled from the story.
The first mosque was built by Muhammad in Medina in 622, right after his migration from Mecca. It was a simple courtyard structure of unbaked brick measuring about 53 by 56 meters, with a shaded portico supported by palm trunks on the north side.
A mihrab is a concave niche set into the qibla wall of a mosque, indicating the direction of Mecca for worshippers. The first known concave mihrab niche was added to the Prophet's Mosque in Medina by Caliph al-Walid I in 706 or 707 and almost immediately became a standard feature of all mosques.
The Umayyad Caliphate, ruling from 661 to 750, introduced the mihrab niche, the horseshoe arch, and the tradition of widening the central aisle in front of the mihrab. Major Umayyad monuments include the Dome of the Rock in Jerusalem and the Great Mosque of Damascus, both notable for their extensive mosaic decoration.
Muqarnas are three-dimensional sculpted motifs created by subdividing vaulting structures into miniature pointed-arch niches, also described as honeycomb or stalactite vaults. The earliest surviving examples include the Arab-Ata Mausoleum (977-978) in Tim in Uzbekistan, the Gunbad-i Qabus (1006-1007) in northeastern Iran, and the Duvazdah Imam Mausoleum (1037-1038) in Yazd.
Mimar Sinan developed a system of centrally symmetric pillars with flanking semi-domes to address the structural challenges of large central domes. His Selimiye Mosque in Edirne (1568-1574), using eight pillars with four diagonal semi-domes, was described by Ottoman sources as surpassing the dome of the Hagia Sophia and is considered the pinnacle of Ottoman domed architecture.
The chahar bagh, also called charbagh, is a formal paradise garden tradition featuring a rectangular irrigated space divided into four equal sections by axial paths meeting at the center. The tradition traces to the Achaemenid Empire; a garden of this type has been identified at Pasargadae, and the form appears at major Islamic sites including the Taj Mahal in Agra, Humayun's Tomb in New Delhi, the Shalimar Gardens in Lahore, and the Alhambra in Granada.