fix: resolve pre-existing TypeScript strictness build errors

- Add @types/node to API devDependencies
- Create HonoEnv type and apply to all routers + auth middleware for
  proper context.get/set("discordId") typing
- Use conditional spreads for exactOptionalPropertyTypes dailyChallenges
  in GameContext, tick engine, and prestige route
- Use conditional spread for optional signature in SaveRequest calls
- Add non-null assertions in shuffle/template index for noUncheckedIndexedAccess
- Cast GameState to never for Prisma InputJsonValue fields
- Exclude vite.config.ts from web tsconfig (it runs in Node context)

apps/api/package.json

@@ -22,6 +22,7 @@
   "devDependencies": {
     "@nhcarrigan/eslint-config": "5.2.0",
     "@nhcarrigan/typescript-config": "4.0.0",
+    "@types/node": "25.3.5",
     "@vitest/coverage-v8": "3.0.8",
     "eslint": "9.22.0",
     "tsx": "4.19.3",