How long and how many generations it has been since the Emperor fked naughty word up
I'd have it be relatively recently if they are going to be in some sort of hidden refuge. If the evildoers are not their semi-immediate family members, or someone they have personal relationships with then the whole having a kin relationship thing isn't terribly relevant unless it is a "only the blood of X line can..." situation. If the Emperor and his colleagues in world destruction are still alive, still ruling the refuge, and the parents, grandparents, uncles, or whatever of our heroes than their own personal responsibility is actually meaningful and their personal relationships with their community have been dramatically effected. If they find out someone in their family did something bad 12 generations back then they probably don't feel much guilt for that, it's just history. Maybe people outside the refuge hate them, but that is just geo-politics if it is some ancient thing, whereas if they have a living association with "the villains" they are grouped with by the outside world that is a lot more interesting.
He destroyed the world when he was fairly young, he is now alive and very old (age depends on race). It has to be just long enough ago that the players' generation could be raised without knowing the outside world.
- Why the players would even consider leaving the safe haven
I would say there is a cataclysm facing the safe haven, or at least this is what they are told. They are sent to get the legendary Macguffin of Plot Contrivance to save it. This isn't actually the main quest, it is just the getting them out in the world quest. It is a simple mission, suitable for first level adventurers, but the elders don't want to do it because they would all be captured or killed on sight for their crimes. They could give the PCs some other reason why it should be them or just offer an extraordinary reward and let the PCs decide to do it, but I like having a cataclysm because then even if a player or two decides their characters are uninterested in the quest for some reason they still have incentive to get out of Dodge.
They should be sent out with very specific instructions meant to keep them from discovering the truth, but they should be of the type that a party (at least as a whole) would never follow. For example they could be told by the high priest not to touch or look at any coins whatsoever lest they be corrupted by the greed of the world. When they get some loot and break this command they notice that all the coins have grandpa's face on them wearing an elaborate crown with "Emperor Da-da-da IV" written around him.
The idea being, fundamentally, that they go out on an errand, and then get swept into a grand quest to save the world and make up for the sins of their forebears.
The Potential Safe Location (is it a hidden town? living in plain sight? underground cavern location?)
I think any Emperor worth his salt who isn't a vampire or something else requiring specific facilities probably picked some place idyllic as his place of exile. Think hidden mountain vale of eternal Spring and/or some sort of vast palace complex. In Xanadu did Kubla Khan/ A stately pleasure-dome decree.
I go two opposite directions on "living in plain sight". On the one hand if they are supposed to not know much about the world or who destroyed it and how then it probably doesn't make sense. Also if we are talking evil emperor and his hench-nobles such characters are probably not willing to just live in obscurity, they would be happier creating some sort of hidden colony to rule over.
However it may make the most sense if you want to give players background creation agency since they could be from different places living ordinary (or extraordinary) lives and just not know who they really are. This would lend itself to some sort of "questgiver who knows the truth summons them all together from disparate places" scenario where mysterious notes direct them to the place where they meet for the first time. You can always just make the Emperor someone who sired a lot of bastards in every corner of the world.
Once again though, I think an actual personal relationship with the Emperor makes it more interesting. Having them be from all over probably doesn't work so well for that.