Concerning the accuracy of the gospel
The introduction to Luke's gospel is in "high" Greek, as was common in historical writing at the time. Moreover, the author claims to have researched events well. We know that the author had gained access to Mark, and claims to have utilized several additional sources until he got "perfect knowledge" of the events that had transpired. Moreover, the author of Luke's gospel often has accurate historical, political, and geographical details that evidences a good deal of research, e.g.:
Luke 3:1-2 Now in the fifteenth year of the reign of Tiberius Caesar, Pontius Pilate being governor of Judea, Herod being tetrarch of Galilee, his brother Philip tetrarch of Iturea and the region of Trachonitis, and Lysanias tetrarch of Abilene, while Annas and Caiaphas were high priests, the word of God came to John the son of Zacharias in the wilderness.
Thus, whoever the author was, he had researched it well. Most scholars believe that Theophilus was a benefactor of the author, who paid for the author to research and write the gospel. Thus he would have had both the time, money, resources, and motivation to get the history right.
Concerning the identity of the author
Internal evidence (evidence from within the Bible itself): The author of this gospel was more concerned with the inclusion of Gentiles into the gospel and stories about Gentiles than Matthew and Mark were. Moreover, the author often needs to explain Hebrew/Aramaic words. This makes it probable that the author was a Gentile (or) writing to Gentiles.
The author is the same as the author of Acts, which has a similar introduction, the same recipient, and the same writing style. The "we"-passages in Acts, i.e. the passages where the author identifies himself as traveling with Paul, indicate that he knew Paul well. Furthermore, these passages give us an indication of when the author was with Paul, and when he was not. In addition, Paul quotes the gospel of Luke (instead of Matthew/Mark) in 1Tim 5:18, indicating that they had significant influence on each other.
The author was well versed in academic Greek, and most likely had a Greek education.
All these facts make Luke a good fit for the author: Luke had an education (Col 4:14), was a Gentile (Col 4:11, 14), was a loyal companion of Paul (2Tim 4:11), and what we know about his travels fits with the "we"-passages in Acts.
If you think through this and look up the verses, you will realize that this is far from overwhelming evidence that Luke wrote the book.
External evidence (evidence from other sources than the Bible): In the immortal words of Wikipedia, "The Church Fathers, witnessed by the Muratorian Canon, Irenaeus (c. 170), Clement of Alexandria, Origen, and Tertullian, held that the Gospel of Luke was written by Luke. The oldest manuscript of the gospel P75 (circa 200) carries the attribution “the Gospel according to Luke”." These church fathers were basing themselves on tradition which in my opinion could hardly have originated after all those who had known the author were dead. One of the inclusion criteria for canonization was that the author was known: at least they thought they knew who wrote Luke. However, though this evidence all points toward Luke, it is possible that the church fathers based themselves on a faulty tradition, which may have been based on an interpretation of the internal evidence as the one I proposed above.
Summa summarum: though Luke is the most likely candidate for authorship, we don't know for sure.
Concerning the anonymity of the gospel
The original recipient of the gospel of Luke would certainly have known who the author was. It is possible that the author's name was not distributed along with the gospel in order to protect the author from being persecuted, or simply because the name of the author was not relevant to the message the gospel contained. None of the other gospel writers felt the need to identify themselves. Or perhaps it was transmitted in some manuscripts and not others, and therefore appears in some of the manuscripts we have, but not others. Or perhaps it was like my post here: though my username is identified along with the post, if you like the content and decide to copy it to some other site/material, you probably won't copy my username as well - it simply isn't relevant. And that's fine - I'm not writing for the credit, and neither was the author of the gospel of Luke.