After travelling around Galatia and Phrygia, Paul came to Ephesus in Asia Minor where Apollos was baptizing in the name of John.
The thesis that the vocabulary of Luke-Acts is special to a physician was deflated by H. Cadbury in his dissertation The Style and Literary Method of Luke (the saying goes that Cadbury earned his doctorate by depriving Luke of his! The argument that the final voyage to Rome is an especially accurate depiction of sea travel can be met with the reply that the author (not Luke) had sailed that way at a later time or appropriated a sailor's account of the same.
The cleavage between the theology of Luke and Paul is simply a consequence of the student going off in his own direction, a venerable tradition.
The disagreements noted between the narrative of Acts and the letters (mainly Galatians) may frequently be reconciled, but in any case are explained if the author of Luke-Acts didn't own any copies of Paul's letters to which he could refer.
It is, after all, improbable that Paul would dispatch a letter both to a church and then to all his sometime companions.
(The "most excellent Theophilus" mentioned in the preface of Luke is most likely his patron, as seen in the similar references to "most excellent X" in the prefaces to the De libris propriis liber of Galenus, the De antiquis oratoribus of Dionysius Halicarnassensis, the Scriptor De Divinatione of Melampus, the Peri ton kata antipatheian kai sumpatheian of Nepualius, and both Josephi vita and Contra Apionem of Josephus.) This Luke has traditionally been identified as the one named in Philemon 24 as a co-worker of Paul.