How Does Catholicism Refute the Reasons Muslims Claim That Jesus Was Not Crucified? Muslims say Jesus didn't claim to be God.

My question is Twofold: 

1) When a Muslim objects to the teaching that Christ was crucified, what is their given reason that anyone ever attempted to crucify Jesus?

2) What proof does the Catholic Church present to refute or nullify those reasons Muslims give that anyone tried to crucify Jesus?
Muslims say that Christians changed the words of Jesus over time, hence inserted false statements into the Bible where Jesus claimed to be God. (See http://www.answering-islam.org/Morin/changed.html) This claim was made long ago, before the discoveries of ancient Bible manuscripts, hymnals, fragments, the Dead Sea Scrolls and other strong evidence that no changes have been made except minor typos and a few passages that may or may not have been in the originals, none of which change any doctrines of Christianity, especially the divinity of Christ and the reality of the crucifixion and resurrection.

Some Muslims hold to the swoon theory (see https://en.wikipedia.org/wiki/Swoon_hypothesis) - that Jesus was still alive when taken down from the Cross, but had passed out and looked dead. The flow of water from Jesus' side described in the Gospel of John when he was stabbed with a spear indicates that he had suffered pericardial effusion (see http://www.compellingtruth.org/blood-water-Jesus.html). You can't survive that. Such a telling detail is strong evidence of eyewitness testimony, not a later invention by someone who had never witnessed a real crucifixion.

As for why people were crucified, the Romans executed thousands of people using this method, including common criminals. Claiming to be a God might upset the Romans, who maintained that Caesar was a God.