Is there any reference that Luke was at the crucifixion of Christ? Or if he had met Jesus face to face? Or was his account of Jesus' life written by research or listening to reliable sources such as Paul?
There are two issues here. The first one is that Luke is not mentioned as an eyewitness to the crucifixion and, being a companion to Paul in the fifties, more than twenty years later, was unlikely to have been an eyewitness. The second issue is that Luke the physician is unlikely to have been the author of the gospel that now bears his name.
Eyewitness to the crucifixion
Mark's Gospel tells us that the women who knew Jesus watched from afar off, with no mention of Luke, or of the disciples
Mark 15:40: There were also women looking on afar off: among whom was Mary Magdalene, and Mary the mother of James the less and of Joses, and Salome.
Matthew's Gospel (Matthew 27:55) agrees with Mark that only women watched from a distance. Luke's Gospel is more ambiguous, allowing that the little group might have included men, but with no mention of names (and certainly no mention of Luke):
Luke 23:49: And all his acquaintance, and the women that followed him from Galilee, stood afar off, beholding these things.
Composition of Luke's Gospel
All the New Testament Gospels were written anonymously. The Gospel now known as Luke's Gospel was attributed to Luke during the second century, apparently on the basis that it appeared to have been written by a gentile, and the same author wrote Acts of the Apostles, which was thought to point to a close acquaintance of Paul as the author.
'Luke' opens the gospel by assuring Theophilus of the veracity of his account. Presumably if he had been present in Jerusalem, or if he had researched the facts about Jesus, or even received his gospel knowledge from Paul, he would have said so, but he does not. If he had learnt it from a reliable witness to the the mission of Jesus, he would also have said so, at the same time declaring this to be clear evidence of the gospel's reliability. Instead, he merely says that the gospel came to him through prior sources, the first of whom was [presumably] an eyewitness:
Luke 1:2: Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word;
It is the almost unanimous consensus of critical scholars that Mark's Gospel was one of Luke's sources, with a strong but less overwhelming consensus that the hypothetical 'Q' document was another of the sources to which he refers in the prologue. John Dominic Crossan says in The Birth of Christianity, page 93, that he spent the 1960s in a monastery, poring over the gospels in parallel columns, word after word and unit after unit, day after day and year after year, studying the scholarly hypothesis and doing source-criticism. In the end he found it absolutely convincing.