A - Thanks for the questions! Scholars might argue about authorship, but we can rest assured that the writers of the Gospels were of the apostolic age, if not actual followers of Jesus (or apostles of the apostles).
The Catechism tells us:
126 We can distinguish three stages in the formation of the Gospels:Tradition holds that Matthew and John were two of the twelve apostles and that Mark and Luke were "apostles of apostles" (Mark of Peter and Luke of Paul).
(1) The life and teaching of Jesus. The Church holds firmly that the four Gospels, "whose historicity she unhesitatingly affirms, faithfully hand on what Jesus, the Son of God, while he lived among men, really did and taught for their eternal salvation, until the day when he was taken up."
(2) The oral tradition. "For, after the ascension of the Lord, the apostles handed on to their hearers what he had said and done, but with that fuller understanding which they, instructed by the glorious events of Christ and enlightened by the Spirit of truth, now enjoyed."(3) The written Gospels. "The sacred authors, in writing the four Gospels, selected certain of the many elements which had been handed on, either orally or already in written form; others they synthesized or explained with an eye to the situation of the churches, the while sustaining the form of preaching, but always in such a fashion that they have told us the honest truth about Jesus."
The Church holds that the tradition would be the norm, unless proven definitively that this is not so. Now, some scholars, esp. starting with German theologians in the 1800's started to question the traditional understanding of authorship based on textual analysis. In other words, they can't prove it was not the four tradition holds to be the authors definitively. Scholarship does has a duty to explore these issues. But, should do so with the eyes of faith. While modern analysis would question John and Matthew more than Luke and Mark at this point, that is missing the point.
Regardless of authorship, the Gospels are authentic representations of Jesus' life, words, ministry and miracles. Early Christians wouldn't have accepted them as true if they were not. We can trust them to be accurately apostolic and authoritative. Authorship (and not "third hand" as accused above) by non-apostles does not mean non-authoritative. Apostolic Tradition is still present in the writings.
Remember that a Muslim will not approach the Gospels as a Christian will. So, don't try to "prove it" to the other person, but rather present the truth in charity and let it stand on it's own. Arguing it to death won't convince another person of the truth of the matter.