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Also, any reason to call it Hoenn Stadium instead of Orre Colosseum (for singles) and Orre Colosseum Doubles (for the existing doubles format)?

Hi! Yes, they are intentional. If two Wob/Wynaut with Leftovers face each other, it creates an endless battle situation, as Leftovers heals more damage than Struggle deals. This rule was part of the official ruleset for Journey Across America 2006, which was Doubles, but just goes to show how important the rule is. In Singles, it is far more important, as Wobbuffet is a very relevant Pokemon. This rule is also present in Gen 3 Ubers as far as I'm aware.

It is in Ubers, but it is not present in the existing Orre Colosseum format, that's why I asked.

The format was originally called Orre Singles way back in the day, but we eventually settled on a new identity for the format. Orre Colosseum is named that way because it is directly based on Orre Colosseum from XD (same exact ruleset), which was Doubles-only. Hoenn Stadium is a nod to the Hoenn games being Singles-focused (whereas the Orre games are Doubles-focused), and to the Stadium games/battle stadium ruleset (bring6pick3 etc.).

You could argue that the leftovers Wob rule should be added to Orre as well, but it's not important as the situation will basically never come up. Also, given that the ruleset there is directly lifted from the in-game Orre Colosseum, it could be seen as an inaccuracy to introduce that rule.