My personal hc is that we're in charge of a sort of Purgatory/Land Between where the near dead/spirited away/lost souls are. As my spirituality professor has said "All afterlives are the same; it's the *people* who make it different." Hence anyone, not just Greeks/Romans/Cretians can access the Hotel. 

Considering the prologue, it's very possible the MC almost died or actually died from some sort of sleep-deprivation- related accident- we *were* in a train station, after all. We could have fallen on the tracks or something.

Everything with The Old Man (who's real name I'm blanking on) could have been in the "waiting room" of the Styx where shades wait for Charon to transport them. The contract allowed us to "bypass" the typical route and go to what is basically where those who have a near-death/spirited away experiences- the Hotel. Either it was set up as is initially or it became that way from an unexpected interaction of different contracts. After all the dead can't leave Hades nor can you punish the dead, but Asterion is obviously*not* in Hades proper since he's being punished. Ergo, we're somewhere in between where the rules of the dead and living intersect- people just pop up like daisies yet leave just as fast, but people who stay can still grow old, have kids, die, ectera. 

"They could either go to the Asphodel Meadows, or the gods might decide that both the MC and Asterion took such heroic actions with the hotel to warrant Elysium. "

I like this thought especially given that there is such emphasis placed on heroic deeds needing witnesses in the trial with Asterion. He only missed out on Elysium in the first place because of this. But now so many have witnessed both his deeds and the MCs. The MC especially, I don't know whether it would traditionally being considered heroic in ancient greek culture but standing up to a god, stopping them from breaching rules of hospitality seems pretty heroic to me.