I believe the great Terry Pratchet summed up the problems of the roman empire something like this (though I could be wrong, its late): You have a good army, decent country, good leadership, but the jerks over the borders keep raiding you, so you conquer them and do some exploiting, not too much or they will rebel, and you will have to conquer them all over again. Then, once the jerks are within the empire, they start complaining about the pricks in the next country raiding them, so you conquer them too, meanwhile the jerks have gone back to doing a little light raiding on the side, then the pricks start complaining about raiding from the next country, the arseholes, so you conquer them too, now the jerks and the arseholes have hated each other forever, this was great when you were warring against them, now there are riots and pogroms in your towns and cities. You have to do a bit more general conquering to keep things under wraps, then the arseholes start complaining about the guys in the next non roman country raiding em, and so on.
The main problems great empires of the past have faced are generally:
1: Even after you conquer them, you still need to keep fairly large garrisons about to keep things under wraps, the romans had to smash up the greeks repeatedly even with large garrisons
2: Where the hell do you stop? Wars are reeeeeally expensive in wealth and men, even back in the day, plunder helped, but that only turned up after the fact and there wasnt any guarantee on the size. So you couldnt just take stuff over all the time, eventually lines had to be drawn and you had to stop somewhere.
3. Communications, and this is the biggy. When the brits first started taking over india, it took 6 months for a message to reach india from britan by ship, so if you had a question going from india, to England to be answered, then back to india, thats 18 months+, obviously this is a problem. If the romans had of been going for world domination, they would have at least needed the telegraph or have had multiple centers of power, which is basically just a pile of separate countries anyway
As for China vs Rome, Id say the romans defiantly had the edge on the battlefield, even if the chinese had explosives, the romans would have gotten hold of them fast, as it stands though china might be on the other side of the moon in terms of moving a full sized invasion force to there from rome with roman era shipping. Id say most likely result is alot of drowned people