I'm most familiar with the Levant, so my answer is fairly specific to that region. Islamization wasn't a particularly even process, and it looked different in other parts of the Middle East.

1. Yes and no. At the beginning of the Islamic conquest the Levant had a number of religious groups, including several varieties of Christians, Jews, Samaritans, and probably a small pagan population. The region was majority Christian, but many would have been considered "heterodox" by the Byzantines. Since you're asking about the entire Middle East, looking farther east there are a number of other religious groups like Zoroastrians, Manicheans, etc.

2. In immediate terms, no. The Islamic conquerors in the 7th century had no expectation — or, arguably, desire — that members of protected (i.e. monotheist, and primarily Christian and Jewish) religious groups would convert. Archaeological evidence shows that existing churches continued to be used and new churches continued to be built during the Early Islamic period. This is, likewise, not limited to the Levant. In eastern Arabia, there was a period of church-building beginning in the late 7th century — after the Islamic conquest — and continuing into the 9th.

The earliest evidence of forced conversion (of protected people) that I know of for the Levant is early 9th century, and even then it's worth noting that this was still explicitly forbidden by the 'Abbasid government. The source I just linked to also makes another point, which is that, even in the 9th century, the largest factor motivating people to convert seems to have been economic hardship, as conversion meant that you no longer had to pay the jizya tax.

Moving forward in time a bit, when the Crusaders entered the southern Levant in the late 11th century, they would have found a fairly mixed population, still. Ronnie Ellenblum argues, in fact, that this is what shaped Crusader settlement in the southern Levant. They settled primarily in majority Christian areas, avoiding places with majority Muslim or mixed Muslim-Jewish populations. It's not really until the later 13th century, after the expulsion of the Crusaders, that prevailing Muslim attitudes toward Christians in the Levant become more hostile, and we can start talking about Christianity becoming "mostly replaced." But even then, this isn't something total and absolute. There are still majority Christian cities today.

3. Indonesia underwent a rather different and — compared to the Middle East — quite late process of Islamization. There, the earliest large-scale conversions take place during the 13th century, and the process continues into the 16th and 17th centuries. Prior to this, the dominant religions were Hinduism and Buddhism.

Generally the Christians at the time in the Middle East were concentrated in West in Judea, some in Syria & the Levant. Although memory doesn't particularly serve right now, I'm sure there where still Coptics in Ethiopia & Egypt. The people elsewhere in the Mid East where still pretty nomadic & even the more civilized where traveling merchants (ala Mohmammad), & with the spread of Islam, they were easily converted and if not they were killed. When Islamic forces moved into the Levant, Syria, Judea etc. they usually force converted, although some did willingly. If you're very interested I'd recommend "The War of the Theee Gods Romans, Persians, & the Rise of Islam" by Peter Crawford. Very good, well researched book.