This is a belated answer, and hopefully one that meets the criteria that R/askHistorians requires. All the information that im drawing from comes from a combination of Pliny the Younger's letter to Trajan (number 97 in the epistules), Suetonius's treatment of Christians (although it is important to note that Suetonius is writing from a distinctly pagan perspective), and James B Rives' "Religion in the Roman Empire" (Which I unfortunately do not have in front of me for citations, so I'm just going off of memory).

In short, yes there are a few generalizations that one can make.

In terms of economic generalizations, one can assume that Christianity was more popular among the poorer classes than the upper classes. This is for a few reasons.

First, Christianity put emphasis on the idea that life is simply a trial for heaven and that life is just testing you. For obvious reasons, the doctrine that "your life may be awful but things will get better" appeals to people who's lives are well awful (side note: I remember reading one author who suggested that the terrible times brought on by the crisis of the third century played a large role in expanding Christianity).

The second reasons is perhaps more interesting: early Christianity did not have limits on who could serve as spiritual leaders. Pliny's letter to Trajan notes

Accordingly, I judged it all the more necessary to find out what the truth was by torturing two female slaves who were called deaconesses.

In short, even slaves could serve as deacons. In a religious system that was exclusively tied to the state (remember that the spiritual leader of Rome was almost exclusively a patrician) Christianity gave the common person some sense of influence.

Last, but most obviously. Early Christianity preached to the downtrodden (Jesus preaching to the whores, Paul's letters etc) it makes intuitive sense that this class would be the most influenced

Because this is askhistorians. I don't feel comfortable answering the ethnicity question because I do not have the sources in front of me. But I do recall Christianity being more popular in the Greek world than in the Latin.