Another problem was that the Romans and the Persians had been at war with each other and had punched themselves out by the time the Arab Muslims were on the move.
Absolutely true, and again, related to the plague. The Iranian people are considered steppe people, although we might tend to picture them as desert-dwelling arab-types. They are quite different. The Sassanid Persians were fighting the Byzantines to a bit of a standstill (that was always the case) and at about that time, plague came in off the steppes and devastated the urban centers.
The Arabs came out of the desert full of Islam, and the Persians were in no shape to stop them, and the Byzantines were in no shape to stop them. The die was cast.