Short answers, pulled from the story.
The early Muslim conquests began in 622 when Muhammad established the first Islamic state at Medina and continued through roughly 750, when the Abbasid Revolution marked the end of the period of rapid centralised expansion. Within that century, Muslim armies extended Islamic rule from the Indian subcontinent to the Iberian Peninsula.
At their height, the caliphates ruled an estimated 13 million square kilometres of territory, stretching from Sind in the east to the Pyrenees in the west.
Both empires were exhausted by decades of war against each other, depleted in manpower, resources, and morale. The Sasanian civil war of 628-632 accelerated political collapse, and recurring outbreaks of bubonic plague impeded conscription of new imperial troops while Arab armies could draw from unaffected nomadic populations.
Khalid ibn al-Walid converted to Islam around 627 and became one of Muhammad's most successful generals. He commanded the Arab force that defeated a major Byzantine expedition at the Battle of Ajnadayn in 634 and is described as the real military leader at the Battle of Yarmuk in 636, which secured Arab control of Syria.
Jews and Christians were classified as People of the Book and generally permitted to maintain their churches and synagogues and keep their own religious organisations in exchange for paying the jizya poll tax. Before mosques were built in Syria, Muslim forces shared Christian churches with local Christians, and numerous churches were repaired and new ones built during the Umayyad era.
Charles Martel led Frankish forces against the Muslim army between the 18th and the 25th of October 732. His assault on the Muslim camp ended with the death of commander Abd al-Rahman Ibn Abd Allah al-Ghafiqi and the Muslim withdrawal at nightfall. Historians note that the Umayyad force had been raiding Aquitaine rather than seeking to conquer France, and that a series of Berber revolts in North Africa and Spain may have played a greater role in halting northern expansion than the battle itself.