"My question posed here, [which] is about the historical context of Luke and Paul's
relationship, and how that relationship might or might not have affected the textual relationships of the writings attributed to them."
Undoubtedly, Paul and Luke had a working relationship - Paul himself says so - and therefore the assumption in note 1 should go unchallenged. I will do my best to look at what we can ascertain about that relationship before considering the relationship of the writings attributed to them, which must also depend, at least in part, on which were written first - Paul's epistles or Luke/Acts.
A hermeneutical approach must also look at the historicity of each reference before considering it 'history'. In particular, in order to establish whether their relationship might or might not have affected the textual relationships of the writings attributed to them, we need to know that the two men were the actual authors of the writings attributed to them. Given the assumptions stated in the question, I accept the obligation in challenging those assumptions to show my working wherever I challenge those assumptions.
Luke in the epistles
Philemon 1:23-24: "There salute thee Epaphras, my fellow-prisoner in Christ Jesus; Marcus, Aristarchus, Demas, Lucas [Luke], my fellow-labourers."
Colossians 4:14: "Luke, the beloved physician, and Demas, greet you."
2 Timothy 4:11: "Only Luke is with me. Take Mark, and bring him with thee: for he is profitable to me for the ministry."
Note: Luke is also mentioned in the subscripts that are included in some Bibles at the end of 2 Corinthians, but this adds nothing to our understanding of his relationship with Paul.
Of the above references, only Philemon is undisputed as an epistle of Paul. Here Paul refers to Luke as a fellow-labourer, which term tells us very little. If Luke was Paul's subordinate, manners may inform Paul to use a term of equality. If Luke was Paul's equal, the term also seems appropriate. If Luke was in a superior relationship to Paul, prudence might require Paul to acknowledge this, but perhaps not necessarily.
Burton L. Mack says, in Who Wrote the New Testament, page 183, that Paul’s letters to the Colossians is not authentic, and that it was written sometime during the 70s. He says the style is different, the vocabulary is different and the rhetoric is different from authentic Pauline letters. In Forged: Writing in the Name of God, page 113, Bart D. Ehrman reports on the analysis of stylistic features in Colossians, conducted some decades ago by Walter Bujard: the kind and frequency of conjunctions, infinitives, participles, relative clauses, strings of genitives, and scores of other things. Everything points to someone with a different writing style from Paul.
The scholarly consensus that Paul did not write the Epistles to Timothy is even stronger than in the case of Colossians. Burton L. Mack says in Who Wrote the New Testament, page 206, that their attribution to Paul is clearly fictional, for their language, style and thought are thoroughly un-Pauline. He says the ‘personal’ references to particular occasions in the lives of Timothy and Paul do not fit with reconstructions of that history taken from the authentic letters of Paul. Bart D. Ehrman writes in Forged: Writing in the Name of God,page 98, about the study undertaken on the pastoral epistles (1 and 2 Timothy, and Titus) by A. N. Harrison in 1921. One of his most cited statistics is that there are 848 different words used in the pastoral letters. Of that number 306, or more than one third, do not occur in any of the other Pauline letters of the New Testament. That is an inordinately high number, especially as about two thirds of these 306 words were used by Christian authors living in the second century.
What this all means is that the only mention of Luke that we have from Paul is in Philemon, and here he merely mentions Luke as a "fellow-labourer." This reference could arguably be used to assert that Luke was not in a superior position to Paul, but this is little more than speculation.
As a postscript, it is worth noting that Luke is not mentioned in Acts of the Apostles.
The Luke/Acts author
Luke, like all the New Testament gospels, was originally anonymous and was attributed to Paul's companion Luke later in the second century, on the basis of inference. Bart D. Ehrman explains in Forged: Writing in the Name of God, page 207, that this author is someone who is especially concerned with the Gentile mission of the early church and who is particularly interested in showing that Gentiles do not have to become Jews in order to be Christian, suggesting that he was probably himself a Gentile. By his occasional use of “we”, the author is claiming to be a travelling companion of Paul. There are three persons in Colossians who were Gentile companions of Paul: Epaphras, Demas, and Luke the physician (Col. 4:12-14). Of these, it seems unlikely that Demas could be the author, since we learn elsewhere that Demas "abandoned" Paul (2 Tim. 2:10). Epaphras appears to have been known as the founder of the church in Colossae (Col. 1:5-7), a church that is never mentioned in Acts. That would be odd if its founder were the author. This leaves one candidate, Luke the Gentile physician. Nevertheless, (p 208), Ehrman says the author seems to be far too poorly informed about Paul's theology and missionary activities to have been someone with first-hand knowledge. Uta Ranke-Heinemann asserts in Putting Away Childish Things, page 7, the Gospel of Luke and Acts of the Apostles do not derive from Luke, the physician and companion of Paul mentioned in the Letter to the Colossians.
'Luke' opens the gospel by assuring Theophilus of the veracity of his account. Presumably if he had 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. It is the almost unanimous consensus of biblical scholars that Mark's Gospel was one of his sources, with a strong but less overwhelming consensus that the hypothetical 'Q' document was another of the sources to which he refers. 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.
Raymond E. Brown says in An Introduction to the New Testament, page 164, there is wide scholarly agreement that Mark was written in the late 60s or just after 70, which would make Luke's Gospel even later. In fact, many scholars date Luke to the last decade of the first century CE, long after Paul's companion would have died.
The first part of the answer to this question must therefore be that the author of Luke's Gospel and Acts of the Apostles was unlikely to have been Paul's companion Luke.
Historicity of Acts
One view is put forward by Sir William Ramsay, who stated "Luke is a historian of the first rank; not merely are his statements of fact trustworthy...this author should be placed along with the very greatest of historians."
Rex Wyler says, in The Jesus Sayings, page 43, that historians consider Acts, written in the 90s, an anonymous work that freely mixes history with legend. Modern scholars tend to be aware of the inconsistencies between Paul's accounts and those of Acts.
Irenaeus, writing around 180-190 CE (Against Heresies III, i, 1), says, “Luke also, the companion of Paul, recorded in a book the Gospel preached by him.” So, as far as Irenaeus was concerned, Luke was the pupil of Paul, recording what Paul had taught.
Relationship of Paul's writings and Luke's writings
If indeed Luke was the author of Acts and the gospel that now bears his name, then we could expect a literary relationship. Irenaeus tells us that Luke wrote a gospel based on what he had heard Paul preaching. However, Irenaeus was wrong, as we now know with some certainty that Luke's Gospel is based on other sources.
When 1 Timothy refers to Luke's Gospel as scripture, this is a reasonable error if 1 Timothy was written in the first half of the second century and Luke was already in circulation. Whether or not Luke was a pupil of Paul, it would be surprising for Paul to refer to the gospel as scripture if he knew it had only been written recently.
Based on the limited reliable evidence available, we can not really say what the relationship between Paul and Luke was, other than that they were Christian missionaries and that they worked together. Paul knew that Luke and Philemon were friends to the extent that they would send greetings to each other.
There is no historical or scriptural evidence that the relationship between Paul and Luke had any influence on what came to be written in Luke/Acts. The testimony of Irenaeus can not be relied on in this instance.