Italy and Egypt are both very close at hand and sitting squarely on supremely strategic territory for the Romans. Egypt both sits on vital lines of communication to the East and (assuming that at some point someone sees the obvious utility of a Suez Canal) one of the most singularly important economic lifelines in the world, especially to an economic giant sitting in Constantinople. I don't see the Romans ever letting the Egyptians pull away without WW1 Ottoman levels of collapse that really don't seem likely to me. If anything they would have huge pressure to integrate them more thoroughly into the wider Roman identity, though I could see some degree of autonomy being maintained.
Italy, while less immediately vital holds the distinction of still being very close, strategically important to exerting power in the Western Mediterranean, potentially economically valuable and an excellent source of manpower. That's not even considering the symbolic value of Rome itself which while less important than it could be, still is of considerable worth.
I could however see a sort of federal system being implemented though, with the Empire as a whole united under the Emperor's authority while maintaining local defense forces. I just don't see them ever being independent and as time goes on, they will likely only be pulled further into the economic orbit of the Heartland and Constantinople in particular.
It also doesn't take much imagination for the Romans to spin a wider Roman identity to supersede general nationalism, assuming nationalism as such even emerges in the same way (though that is certainly possible). Particularly given how well educated the Roman population is for its time, and how well centralized, along with shared history of all its constituents, and shared enemies.
For population, I could see it going either way but in the long run, the region will have been run by an efficient and centralized government for much longer, the peace kept far more efficiently and will have had a far more active and dominant economic powerhouse to serve as foundation for later population growth. That's a solid foundation both for urbanization and industrialization once the resource issues do eventually get sorted out likely by importation of the necessary resources from a number of their neighbors. On top of that, they will have an extremely well educated population for the time, again furthering economic growth and it is only a matter of time at that point before they either make or adopt more advanced agricultural methods. Combined with access to two excellent breadbaskets I firmly believe that by the time the second industrial revolution rolls around they will be on par at least with any single great power.
See I just don't see the Romans being passive in their dealings with their neighbors. All their experience will tell them that the Army is vitally important, and that they would much prefer to do their fighting in their allies' lands rather than their own. Again. Further, territories like Scythia are not just nice to have, they are a matter of life and death to Constantinople and the heavily urbanized heartland. Especially before the rise of modern agriculture, but even then they will likely still need food imports and I cannot see them being able to afford to be passive. Especially since Scythia is a natural spark point for conflict given how badly any potential Russian power is going to want it, and how many resources Rome is likely going to be importing from Russia given the relatively easy river travel (wasn't there a canal started between the rivers there to facilitate trade to the Black Sea from Russia?)
The first wave will certainly be a bit troublesome, but I don't think catastrophically so. Oil though you are right will be absolutely key. Might even eventually be motivation to take Mesopotamia itself around that time to secure access to those fields, though they will certainly have plenty to start off between Vlachia and Georgia. Scythia has some as well as I recall, and they will likely be able to import a great deal of resources from the east.
That economic activity between East and West through a potential Suez, along with Constantinople being a major economic powerhouse are the two things that, properly leveraged, could help offset lack of access to coal to a certain extent.
You're right that in light of this the Navy and Air Force absolutely have to be priorities, I just think there is more than enough economic power in the Empire and its colonies in the East that it doesn't have to be to the detriment of the Army so much that it is at a disadvantage against any single European power like France or Germany. Of course things get hairy if they get ganged up on but that is true for any great power even if it is a bit more so for this Roman Empire. That is the job of the Diplomats, to do everything in their power to make sure that you don't have to fight a two front war, and that if the worst happens and you do, you have plenty of allies to help.
That was a lot more than I meant to write.