conversions, conquests, etc.

I really think these ought to be disaggregated. Certainly the conquests were rapid, particularly after the death of Muhammad, but conversion was a much slower process. Our best estimates are that it probably took a couple hundred years for Islam to predominate in the conquered territories, and large religious minority populations still exist throughout the region, in many cases only truly dwindling in the 20th century (as in the case of Turkey.)

On the conquests, I think this thread more or less covers the basics: https://www.reddit.com/r/AskHistorians/comments/4libs1/were_the_muslimarab_conquests_of_the_8th_century/

On Christianity in Arabia, it’s difficult to say exactly in what proportions they spread, but across the peninsula as a whole, I think you would have to conclude that Christianity (and Judaism) did in fact spread widely and deeply, with Judaism being predominant in Yemen (which was actually a Jewish kingdom, the Himyarites) and Christianity apparently being the majority religion in Northern Arabia and having a presence in other parts of the peninsula as well.