There's a person in the village named Leon. He's the one who built the teleporters to the beach and hot springs. I've never talked to him myself, but he sounds like he might be a good lead.

[Isaac also comes to mind, but that's not a person Ginia is comfortable sending Red to seek out. Not that she thinks Red would come to any harm, but there are deals best not made.]

And as for your question about the Battle Dome, I fight people and Third Party cultists.