I'd personally also think of many regionalist parties seeking independence, maybe even controlling some provincial governments.
It is likely that the empire would have expanded beyond the Rhine and Danube; Germania would be a vassal state that, if drawn into the Empire would have strong elemts of regionalism. Caledonia too would have been subdued eventually - by trade if not by war; I expect Caledonia to also exhibit regional autonomous traits for example. Other that that there would be a 'uniformity' across the empire with divisions based on geography and the odd cultural split.
Ethnically the empire would be mixed, but not more so than Europe is today - just that it would have happened across a longer timescale. For example, Meditteraneans, Nordics etc would still be the dominant race in their hartlands, but there would be a mix - then again there may have been less of a mix than occured in reality after the fall of the Empire (UK is a case in point)
I prefer to pretend Christianity becomes no more than another 'cult' among the many and that there is no east/west split.
The Empire could probably only function as a heavily federalised state; a cross between the current EU and the USA
Excuse the source.