My comment that was linked by /u/i_like_jam answers part of your question, but there is definitely more material to discuss. As I am personally fascinated by this, I hope you enjoy this massive wall of text!

The main reason why this question is complicated is that the answer is different for different sorts of Christians in different places. Early Islam was fairly inclusive of Christians and Jews, and there is even evidence that these people also fought alongside the Arabs following Muhammad's teachings. For Christians within Arabia, I'm inclined to say that it wasn't a particularly threatening movement. Muhammad talked about ending paganism, not overturning the status quo of a Near East dominated by monotheism already, and perhaps he couldn't have done so even if he wanted to, as Arabia was a part of the late-antique world and Christianity had long been entrenched within the peninsula. To illustrate this point, an earlier attempt by a Jewish king of Yemen to persecute Christians in his domain in the sixth century led to a Byzantine-backed Ethiopian invasion of Arabia to remove him.

Of course there were tribal conflicts and personal disagreements that did create a lot of conflict, but Islam was not intrinsically threatening to Christians. There were also theological differences, such as Islam's denial of the Trinity and Jesus' divinity, but I feel that these differences disguise how the rise of Islam was not entirely the rise of an 'Other', but the coming to power of a movement that was aware of its context and itself a product of the late-antique world. The majority of our sources for instance were written by ecclesiastical writers, who often damned Islam as a heresy and the Arabs as barbarians; this I think was not representative of the attitude of the 'average' Christian, who no doubt found ways to deal with their changed circumstances without going around fulminating against the regime of the vile Saracens. If we look at the evidence more critically, a different (and perhaps a more interesting) picture emerges. Basically, the opinions of contemporary sources only represent the views of a narrow section of society, whilst the views of Christians as a whole are probably not only unrecoverable but also I think not a particularly good way to look at things, simply because of how diverse Christian society was in the seventh century.

Firstly, I think it would be wrong to use the term 'Catholic Church' here, as it has the modern connotation of a Church associated with and dominated by the influence of the papacy, when in the early seventh century the papacy was still an institution under Byzantine control, though from the 630s onwards there was a concerted attempt by the papacy to assert its political independence. I think it is better to think of Christianity as a faith of many different Churches in this period. The Visigothic and Frankish Churches for example were fiercely independent and possessed a great deal of secular influence, whilst the Anglo-Saxons were extremely reverential of the papacy for its role in converting them. Even Italy was divided, for the region around Verona was in schism with the papacy over the Three Chapters Controversy from the 550s, and Ravenna had sought to escape papal authority through secular patronage for decades.

Byzantine Christians meanwhile were divided into two camps, Chalcedonian Christians who followed the doctrines confirmed at the Council of Chalcedon in 451 (which was also largely the view of Christians in the west), and Miaphysite Christians who did not and who formed the majority of the population in Syria and Egypt. The imperial government in the 630s also promoted Monotheletism, a doctrine that was perhaps a natural extension of past imperial gambits to reunite the divided Christian communities within the empire, but only ended up causing more problems as radicals on both sides raged against this new heresy. There were also conflicts between different patriarchates over jurisdiction and also over simple personal enmities. The Church in the seventh century was a broad coalition of diverse institutions that subscribed to similar doctrines but were themselves products of their own contexts. These conflicts continued even amidst the Arab Conquests and shaped the world that emerged afterwards, something I don't think historians have explored enough.

Now that I've got that out of the way, how did Christians respond to the rise of Islam? The most immediate sources are several Syriac chronicles/fragments written down by Palestinian monks in the 630s/640s (which I've quoted here), but they only really talk about how the Arabs invaded Palestine and enslaved a lot of people. Excessive hyperbole is pretty standard for ecclesiastical writers at the time, so I wouldn't treat this as a particularly accurate account of what local Christians thought. Another contemporary source is a sermon by Patriarch Sophronius of Jerusalem, who lamented to his congregation about the fact that they weren't able to travel to Bethlehem in 634 to celebrate the Nativity due to Arab raiders. However, the way he characterised the Arabs as barbarous and generally unpleasant had a long and proud history in the Roman Empire, as the Romans were never fans of anyone who didn't live within the empire, so this does not necessarily mean that Sophronius was angry at the Arab raiders because of their religion.

Generally, many Christians in the Near East were happy to co-operate with their new rulers, as we basically only have one instance of a Christian resistance movement in occupied regions and frankly it sounds more like Byzantine special forces inserted into Syria rather than a native movement. Islam was definitely not a religious movement out to get Christianity and we have records of defectors everywhere, particularly in Egypt where the Chronicle of John of Nikiu and papyrological evidence show that many Byzantine officials, even provincial governors, remained in place after the Conquest. In Syria, the father and grandfather of the eighth-century theologian John of Damascus also served the new Arab regime, despite their services to the Byzantine state before the Conquest. Ironically, John of Damascus was one of the first theologians to write a detailed thesis arguing that Islam was a Christian heresy. However, the fact that John was able to write about the Muslims in such a way in Syria tells us a lot about how Christians were able to find a modus vivendi with the Arabs in reality. The same was true for John of Nikiu writing in Egypt a few decades earlier, as he damned Monothelete heretics and corrupt Byzantine governors throughout his Chronicle, but was on the whole rather fair to the Muslims, particularly their leader 'Amr. I do get the impression that he wanted to be ruled by Christians instead, but that did not mean that he was particularly upset about his present situation. It wasn't ideal, but it was something that he had to live with.

At this point it is also worth pointing out that many Arabs lived within the empire already, so one useful way of thinking about this is to think of the Conquests as an internal insurrection by a people well-connected to the Byzantine world and keen to seize its wealth rather than to destroy it. Combined with the lack of destruction layers in urban sites, many historians dismiss the idea that the Arab Conquests were exceptionally violent. It was still a war of conquest, but its effects were exaggerated by contemporary Christians and indeed even now thanks to how politicised Islamic history has become. The Near East had also been occupied for Persian forces for more than a decade between the 610s and 628, so if anything, being ruled by the Byzantines was stranger than to be ruled by the Arabs, especially as the Byzantines ordered the forced conversion of Jews and persecuted Miaphysites in the brief time between the Persian War and Arab Conquests.

Check out shlin28's excellent overview here. He discusses Roman perceptions of Islam and answers many of your questions.