How has the Tevinter Imperium lasted so long?

That's true, without the slave workforce their economy and their army would collapse. If every slave just downed tools the country would grind to a halt. The biggest threat to them does come from within if either the Qun, Solas or anyone else manages to infiltrate the slave class sufficiently to undermine the state. Hessarian overcame the problem during his time by converting to the faith of the person who had opposed slavery and allowing large numbers of the predominantly elven slave workforce to leave the country if they wished. As the account of the Long Walk states, many of them turned back for Tevinter and slavery when they realised how tough it was outside the Imperium.

Since then they have made up the difference in human slaves and racial prejudice even among the slaves seems to keep them from presenting a united front. Also, as Dorian points out, some slaves are treated rather well, particularly those household slaves directly serving the family, and there is as much a hierarchy among slaves as there is in the free population, so those who do well under the system have no incentive to want to change things. Also poorer families see selling themselves or their children into slavery as a way out of poverty. Other countries also tacitly support the system by allowing the slave trade to operate, even though officially it is outlawed.

If anyone did find a way to break the system of slavery then the destruction of Tevinter would likely follow.

The Roman Empire lasted for a thousand years in the east after the barbarians conquered Italy.

I imagine the 'strength' of Tevinter's slavery is more akin to dependence at this stage. They don't have the economy to clean out slavery and replace it with a labour class but that's just me spitballing and I'm not an economist so I'm not even sure how slavery vs laborers would work, maybe they make up for it by buying stuff yay capitalism. There's also the fact that it wouldn't be just about economic realities in Tevinter, it'd be about blood too. 

Anyway, it's magic right? They're the only country to embrace it without reservation. They've been practicing a kind of unofficial eugenics program for ages and power and influence is fundamentally tied with magic, presumably that extends to things like ingenuity and skill and raw destructive capability as well. It is, for me, probably the most important part of what I wanna see in Tevinter, magic as a way of life, magic as something to be proud of, unrestrained, glorious magic. I'd split the Magisters influence 50/50, partly responsible for Tevinter's decline, partly responsible for it still being around. In the rest of Thedas, when you trace the ruling dynasty's lineage back far enough, what you'd find is an exceptional person. Someone more worthy than the rest generally because of some deed or happenstance. That isn't true in Tevinter really, there's a qualitative difference with the ruling class there that you can't emulate. You are a mage or you are not. In contemporary Tevinter this seems to have resulted in the kind of entitlement where rulership is almost more a point of academic pride, divorced from the realities of and duties of leadership. Hence holding their buildings together with magic. In a resurgent Tevinter with Magisters looking beyond their personal squabbles and ambitions it could be magical architecture, glamorous, involved. Instead it's necessary magical maintenance. On the flipside, I imagine this reflexive "MINE!" attitude coupled with an awareness of how far they've declined and how precipitous their current position is results in an extremely potent need to defend what they have left. Cuz it isn't just about the land, it's about identity. Without Tevinter they aren't the all mighty mage lords anymore. What's the closest comparison point? Their southern brethren living on the largess of the mundanes? Or the mutilated, self-loathing saarebas? I'd guess that a fat chunk of the magistocracy would choose glorious Braveheart death before either of those options. 

As others have already pointed out, they didn't really survive the 1st Blight + associated aftermath. They had a lot to lose and they lost most of it. It just so happens that 25% of the most powerful empire in the world is still decently hefty in its own right. They had a technological/societal headstart on the other societies and when they others caught up Tevinter had recouped enough that taking it down wasn't a casual affair (not that they didn't try). 

I can't help but think of late Byzantine empire when I look at current Tevinter, real close to being knocked down by the Ottomans. Just need an errant crusade. But I also hope that DA4 will be their opportunity for a Komnenian restoration which is mixing my history but w/e. 

There is the generally accepted and well argued history truism that one of the reasons the US institution of slavery became more unpopular in the years prior to civil war was that it was super expensive and crippled the southern economy, and even if the south won the war (the premise of that HBO show game of thrones show runners wanted) it would have been untenable to maintain slavery.

this truism largely is because of America’s industrialization and the rise of free market capitalism in the 1800s so Tevinter being more like a renaissance era European economy, the economies that thrived off various forms of slavery, they likely are not facing the consumer based economic forces that made the idea of having an unpaid labor force who could not turn around and spend their wages on goods and services an economically bad idea.

but in general the free markets of post slavery US made iconic US slavery not economically viable but also made quite a bit of money off colonial projects exploiting (sometimes to the extreme of slavery) people across the world. Colonialism was a thing long after the civil war in America. 

So unfortunately when greedy bastards want power and money owning other people and profiting from their toil is historically profitable for short term gains. King Leopold of Belgium made a lot of money off colonialism style slavery in the Congo. But in the long term, the crap he pulled there was economically devastating, at least for the people of the Congo, Belgium is pretty wealthy.

Is Minrathous an original city or did it always exist as an abandoned elven city that the human tribals decided to settle in and remodeled to their liking.