So if it somehow did get said social pressures, would it still be an Empire?
I don't see why it wouldn't, but the nature of the Empire would change a great deal I'd think, but the most destructive elements of the old empire would be dialed up to 11.
Rome was more or less screwed by massive public subsidy schemes and uncontrolled immigration by germanic and Persian tribes and outsourcing their military to said tribes...The West died in the mother of all welfare riots. This was done to maintain political monopolies...in an industrialized society the incentives to abuse such a system is even more abundant and the US largely avoided the fate of say the EU, because the dawn of its industrial age occurred concurrently with a massive cultural shift towards self reliance and a near religious obsession with self governance.
Rome doesn't have that in post Republic era and it had a bastard version of that when it did.
Would its technological advances get “stolen” or copied by foreign enemies before they conquer them?
I don't think so...the only nation that comes close to even understanding the basic principles behind say..a steel factory would be China and maaaybbbeee a couple Indian kingdoms...And neither of them really did anything with that.
How would social mobility be? Would there still be a Roman Emperor? How long would it take to abandon slavery?
Given how Romans viewed Actors and merchants...One imagines the next massive social insurrection will be the performers followed by the business class.
How that affects Rome depends..Slavery lasts as long it takes a Roman tax collector to put two and two together about the benefits of being able to shake down a working class that gets decent wages.
Rome would still have an Emperor..or several. But the extent of their executive powers would be different I'd imagine.