Improve UX with merged run view, method badges, grouped encounters, and mobile nav

Merges the run dashboard into the encounters page as a unified view at /runs/:runId,
adds encounter method grouping in the modal and badges on route rows, improves the
home page with recent runs, adds mobile hamburger nav, and polishes empty states.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

backend/src/app/schemas/game.py

@@ -7,6 +7,7 @@ class RouteResponse(CamelModel):
     game_id: int
     order: int
     parent_route_id: int | None = None
+    encounter_methods: list[str] = []
 
 
 class GameResponse(CamelModel):