Were Jews and Christians ruling the world when Mohammed was alive?

Answer 1

No.

Jews: Jews did not have any nation or state since the destruction of the Hasmonean Empire in 37 BCE and were a persecuted minority in the Byzantine Empire and the Sassanian Empire.

Christians: As for Christians, although the Byzantine Empire was a nominally Christian State, it was really dominated by pagans. Christians formed roughly 10% of the population of the Byzantine Empire even as late as the 600s CE. Christianity would not become the dominant religion in Europe until the mid-900s.

The Chinese Sui and Tang Dynasties had around 45-50 million people compared to the Byzantine Empire with roughly 17 million people. Even the Sassanian Persian Empire with 22 million people was larger than the Byzantine Empire. Neither of these states (China or Persia) had any significant minority (greater than 3%) of Christians. In fact, China would not receive Christian emissaries in any significant number until the mid-1200s with the Pax Mongolica and the Silk Road. Additionally, the Byzantine Empire was equally matched in strength and ferocity by the Sassanian Empire. It is actually because these two countries had been going to war for centuries and whittling away their armies that the Muslim conquerors (led by Khaled ibn-Walid) were able to destroy Sassanian Persia and conquer over two-thirds of the Byzantine Emprire. Reza Aslan, a noted Persian-Amerian academic has noted as much. The Sui Dynasty was probably the most powerful nation on earth, fighting a strong war of conquest against the Korean Empire of Goguryeo - which eventually failed through attrition on the Korean side.