.. Why did they build a copy of the Underworld with real lava?

Ooo, good question, and not really answerable. The play's producers must have a hell of a budget.

The real-world answer is that it makes the game a whole lot more fun. In Nancy's world, the world of this game, it makes not the slightest, faintest whiff of sense for a tiny four-person theater troupe to construct working models of Hades complete with flames, obscure locks, and acid pits that are supposedly going to be lifted into place during a short play, especially given that it's to do with ancient Greek mythology and ancient Greek plays didn't use sets, costumes, or even makeup. These sets would be more believable if they were constructing a Hades theme park, and even then it would have to be NC-17, given the potential for actual death.

But let's not let logic or reality ruin the fun, shall we?