You still avoid my questions. What evidence do you have to attribute the Gospel of Luke to Dr. Luke? None. You simply infer that Luke was written by a companion of Paul. There is no strong basis for that inference. Possible? Yes. But equally possible otherwise.

Regarding the "We Document", there are far more likely explanations than yours. Scholars who read the original Greek recognize instantly that the author of the first part of Acts (that talks about "they") and the second part (the "We Document") were not one and the same. The style and language are completely different. They were later severely edited and combined into one document to serve the purposes of Orthodox editors. Much more on this later.