Was the Luke of Colossians the author of Luke/Acts?
Probably. As the two volumes do not themselves include the author's name, we can't be sure that the author was named Luke at all. However, Luke is only mentioned 3 times in Paul's letters and there is no indication there that he was a particularly prominent personage. Therefore, any external evidence for Luke as author is somewhat stronger than it would be if Luke were already recognized as an authority in the church.
According to Wikipedia, "Irenaeus, Tertullian, Clement of Alexandria, Origen, and the Muratorian Canon all regarded Luke as the author of the Luke-Acts" Also, P75 clearly labels the end of the Gospel with Λουκᾶς:
One reason Luke might have been assigned authorship, however, is Paul's reference to him as a physician, which implied a man of great learning. (Galen, for instance, was known both as a medical doctor and as a philosopher. Intellectual typecasting was less common in those days.) Since both Luke and Acts are masterfully written histories in the classic style, it might have been tempting to make that connection. However, it does seem unlikely that no other names would have been mentioned if the authorship were simply an educated guess. (Compare with the suggestions for who wrote Hebrew to see the significance.)
One intriguing hint that the Luke of Colossians 4:14 is the author of Luke is the word choice in the "eye of the needle" saying. Rather than ῥαφίς, which refers to a sewing needle, Luke uses βελόνη, which apparently refers to a surgical needle. A medical doctor of the time would have had a wide variety of tools for dealing with injuries and may very well have imagined preparing to close a wound rather than preparing to fix a garment when he heard the saying.
Does it matter to interpretation that Luke was a doctor?
This is a far thornier question. When it comes to understanding Luke 22:36-38, I tend to agree with you: there's very little reason to bring up Luke's profession at all. But in the broad sense, I think it does matter. In fact, I think knowing something about the author matters a great deal when it comes to interpreting any text. Knowing who an author is can aid in understanding what he wrote and why. While we should not be slave to authorial intent (especially when we can only speculate about it), we should be guided by it (especially when recorded in the work itself).
Thankfully, we know what the author of Luke/Acts was trying to accomplish:
Inasmuch as many have undertaken to compile a narrative...it seemed good to me also...to write an orderly account for you...that you may have certainty concerning the things you have been taught.—elision of Luke 1:1-4 (ESV)
In other words, Luke was acting primarily as a historian (commissioned by Theophilus) in his writing. We must always evaluate his work in those terms first.
In broad terms, the care and attention to detail required of a doctor would have served Luke well as a historian. This is born out in Luke's meticulous usage of official titles. When we compare his output to Mark (who seems to mis-remember minor details) or Josephus (who displays considerable bias), we find Luke a man of even temperament and thoughtful care. These are evident even if we did not know or suspect that the author was a doctor, but knowing that detail fills in another piece of the puzzle.