The degree and/or level of sophisticated intellectual discourse of those "Roman-Times" 'Philosophers of God' and metaphyscis afficinados.
Well a good many the Fathers, with a couple of notable distinctions, were heavyweight Platonists in their own right. Justin, the first of that era, argued that if Revelation was true, it was arguable philosophically. It's from the traditional black garb of the Greek philosopher were get our ecclesial black today.
There would have the equivalent to a "Popular Mechanic" journal read by armchair thinkers everywhere. No?
We called them 'gnostics'
What was the mind set and philosophical catch-phrases and talking points of the time?
Probably much like today. At the height of the Arian dispute the popular chant of the Arian party was 'there was a time when he was not' — it's a rather snappy ditty in Greek, but I can't remember it.
There is a source who notes in a contemporary journal his despair at not even being able to buy vegetables in the market without being accosted by stall-holders who want to dispute whether Christ is eternal or created!
The mere fact that conversions to Christian thought [that reaches it's apex in accepting Christ as Redeemer] ---IMHO, may NOT have been a giant leap of faith ---it had to be presented & debated over large expanses of the known world of the time ---a public debate etc etc . . . for the apostles of Jesus to 'gather the fish' ... and moreover, 'make fishers of men'.
I'm not so sure.
Before the church was theological, it was liturgical. Being a Christian meant initiation into the Christian Mysteries. Theology only developed subsequently, 'faith seeking understanding' as Anselm defines it, not understanding seeking faith.
Many would be seekers, looking for understanding, sure, and would be drawn to the answers the church was preaching — Paul's discourse with the Athenians in Acts 17 is such a case — but those answers are received in faith, the same today as then.
There's big debates over Christology for example, over the person and nature of Christ ... but the resurrection was not an issue. You either believed He rose again or you didn't, and if you didn't, you didn't become a Christian and then argue the details, you just diodn't become a Christioan. As Paul argues, it's absolutely fundamental. If you don't believe in the resurrection, then being a Christian is rather pointless.
How to plead your case to heathen Roman citisens? What did it take? How did one "Prep" oneself?
Well, the Mysteries aside (as stated above), that's where theology evolved. Luke's Gospel, for example, is the gospel of social justice. This was a huge attraction. It accorded everyone equal rights as a person before God, whereas Roman law saw only male Roman citizens as 'a person' and everyone else peripheral to that.
Matthew addresses a Jewish sensibility. Mark a populist sensibility (the pace of Mark is breath-taking). Luke addresses an educated Greek sensibility. John a primarily Jewish spiritual sensibility.
In the 60s, there were running street battles in Rome between rival Christian and Jewish mobs. (That's why Nero thought he could scapegoat the Christians, they could be a noisome lot!)
By the middle of the next century, the Church in Rome had an extended social outreach programme, caring for widows, orphans, the sick and the dispossessed. A Roman senator is recorded as complaining to his brothers in the Senate that 'even the Christians care better for their people than we do. Documentary evidence shows some 1,500 people were listed 'on the books' in this care programme.