1. When a Muslim objects to the teaching that Christ was crucified, what is their given reason that anyone ever attempted to crucify Jesus?
I interpret this question to mean "What does Islam teach about why the Jews wanted to crucify Jesus?" and I will attempt to address that. If I have misinterpreted OP's intent then please let me know.
In short, the Qur'an says that the Jews wanted to kill Jesus for the same reasons they abused the prophets who preceded Him: for exposing the wickedness of the Jews and calling them to repentance. In this, at least, the Qur'an is somewhat in agreement with the Bible.
Surah 3 (Ali 'Imran) ayah 112 says (emphasis mine):
ضُرِبَتْ عَلَيْهِمُ الذِّلَّةُ أَيْنَ مَا ثُقِفُوا إِلَّا بِحَبْلٍ مِّنَ اللَّهِ وَحَبْلٍ مِّنَ النَّاسِ وَبَاءُوا بِغَضَبٍ مِّنَ اللَّهِ وَضُرِبَتْ عَلَيْهِمُ الْمَسْكَنَةُ ۚ ذَٰلِكَ بِأَنَّهُمْ كَانُوا يَكْفُرُونَ بِآيَاتِ اللَّهِ وَيَقْتُلُونَ الْأَنبِيَاءَ بِغَيْرِ حَقٍّ ۚ ذَٰلِكَ بِمَا عَصَوا وَّكَانُوا يَعْتَدُونَ
"They have been put under humiliation [by Allah ] wherever they are overtaken, except for a covenant from Allah and a rope from the Muslims. And they have drawn upon themselves anger from Allah and have been put under destitution. That is because they disbelieved in the verses of Allah and killed the prophets without right. That is because they disobeyed and [habitually] transgressed."
Surah 4 (An-Nisa) ayah 155 says (emphasis mine):
فَبِمَا نَقْضِهِم مِّيثَاقَهُمْ وَكُفْرِهِم بِآيَاتِ اللَّهِ وَقَتْلِهِمُ الْأَنبِيَاءَ بِغَيْرِ حَقٍّ وَقَوْلِهِمْ قُلُوبُنَا غُلْفٌ ۚ بَلْ طَبَعَ اللَّهُ عَلَيْهَا بِكُفْرِهِمْ فَلَا يُؤْمِنُونَ إِلَّا قَلِيلًا
And [We cursed them] for their breaking of the covenant and their disbelief in the signs of Allah and their killing of the prophets without right and their saying, "Our hearts are wrapped". Rather, Allah has sealed them because of their disbelief, so they believe not, except for a few.
2. What proof does the Catholic Church present to refute or nullify those reasons Muslims give that anyone tried to crucify Jesus?
I interpret this question to mean "What approach does the Catholic Church recommend employing to refute Islamic views of the Passion, namely that Christ was not crucified?" and I will attempt to address that. If I have misinterpreted OP's intent then please let me know.
Paul's answer is correct in that Muslims believe that the Gospel, as it exists now, has been distorted and/or corrupted by early Christians removing important passages and adding in things that weren't true. For this reason alone, a Muslim is highly unlikely to accept any argument based in scripture or in the Church's teaching (which is based on scripture). Conversely, they believe that the Qur'an has been kept free of any such corruptions since it was first revealed to Muhammad. Thus, in order for a Muslim to consider it "valid", a direct refutation of the claim that Christ did not die on the cross would need to have its basis in the Qur'an itself. This presents a formidable task for any apologist because not only does it require them to have at least some understanding of the Qur'an -- which most Christians have probably never read -- but also functionally requires a Muslim to accept at least one of the following statements as true:
The Qur'an is subject to the same inaccuracies and distortions that the Bible and Torah are accused of having.
The Qur'an is unclear, incomplete, or self-contradictory (even after accounting for the belief that verses in the Qur'an can be "abrogated" by other verses, which is another very deep and complicated topic).
Muhammad did not faithfully convey the Qur'an as it was revealed to him by God; he added or omitted things as he saw fit.
Muhammad never received any divine revelation: the Qur'an and Islam are lies.
But their rejection of Jesus' death and resurrection goes deeper than the belief that the Bible is not an accurate record of the life and teachings of Christ.
The Qur'an explicitly denies that Jesus died on the cross in surah 4 (An-Nisa), ayat 157-158, claiming that someone else (usually believed to be one of the Disciples) was somehow made to resemble Jesus and was crucified in Christ's place:
وَقَوْلِهِمْ إِنَّا قَتَلْنَا الْمَسِيحَ عِيسَى ابْنَ مَرْيَمَ رَسُولَ اللَّهِ وَمَا قَتَلُوهُ وَمَا صَلَبُوهُ وَلَٰكِن شُبِّهَ لَهُمْ ۚ وَإِنَّ الَّذِينَ اخْتَلَفُوا فِيهِ لَفِي شَكٍّ مِّنْهُ ۚ مَا لَهُم بِهِ مِنْ عِلْمٍ إِلَّا اتِّبَاعَ الظَّنِّ ۚ وَمَا قَتَلُوهُ يَقِينًا
بَل رَّفَعَهُ اللَّهُ إِلَيْهِ ۚ وَكَانَ اللَّهُ عَزِيزًا حَكِيمًا
"And [for] their saying, "Indeed, we have killed the Messiah, Jesus, the son of Mary, the messenger of Allah ." And they did not kill him, nor did they crucify him; but [another] was made to resemble him to them. And indeed, those who differ over it are in doubt about it. They have no knowledge of it except the following of assumption. And they did not kill him, for certain."
"Rather, Allah raised him to Himself. And ever is Allah Exalted in Might and Wise."
On a deeper level, Muslims believe that God would never allow one of His chosen messengers to be humiliated in such a way; crucifying Jesus would have been tantamount to sinful men triumphing over God (see this Wikipedia page for a more detailed explanation).
How do we refute the Qur'anic narrative that Jesus was never crucified?
A good starting point is this:
The Qur'an says, in surah 61 (As-Saf) ayah 14 that God supported "those who believed" (i.e. as the Apostles believed) and made their belief the predominant mode of Christianity. There is a wealth of historical evidence that belief in the Trinity and the Resurrection (in the form that we know those doctrines today) originated as early as 110 AD, and were formally adopted in the first Council of Nicaea in 325 AD and the First Council of Constantinople in 381 AD -- more than two centuries before the first verses of the Qur'an were (allegedly) revealed to Muhammad. This is indisputable historical fact.
If we accept Surah 61:14, then God explicitly claims responsibility for the dominance of Trinitarianism, which is decried throughout the Qur'an as "shirk" ("polytheism"), which God abhors! Does this mean that God Himself intentionally and deliberately led believers astray?!