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/web/src/context/GameContext.tsx

@@ -174,7 +174,7 @@ export const GameProvider = ({ children }: { children: React.ReactNode }): React
         if (stateRef.current && !isSyncingRef.current) {
           void saveGame({
             state: stateRef.current,
-            signature: signatureRef.current ?? undefined,
+            ...(signatureRef.current !== null ? { signature: signatureRef.current } : {}),
           }).then((response) => {
             setLastSavedAt(response.savedAt);
             if (response.signature) {
@@ -222,7 +222,7 @@ export const GameProvider = ({ children }: { children: React.ReactNode }): React
     try {
       const response = await saveGame({
         state: stateRef.current,
-        signature: signatureRef.current ?? undefined,
+        ...(signatureRef.current !== null ? { signature: signatureRef.current } : {}),
       });
       setSyncError(null);
       setLastSavedAt(response.savedAt);
@@ -267,7 +267,7 @@ export const GameProvider = ({ children }: { children: React.ReactNode }): React
           totalGoldEarned: prev.player.totalGoldEarned + clickPower,
           totalClicks: prev.player.totalClicks + 1,
         },
-        dailyChallenges: updatedDailyChallenges,
+        ...(updatedDailyChallenges !== undefined ? { dailyChallenges: updatedDailyChallenges } : {}),
       };
     });
   }, []);