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/src/middleware/auth.ts

@@ -1,7 +1,8 @@
-import type { Context, Next } from "hono";
+import type { MiddlewareHandler } from "hono";
+import type { HonoEnv } from "../types/hono.js";
 import { verifyToken } from "../services/jwt.js";
 
-export const authMiddleware = async (context: Context, next: Next): Promise<void> => {
+export const authMiddleware: MiddlewareHandler<HonoEnv> = async (context, next) => {
   const authorization = context.req.header("Authorization");
 
   if (!authorization?.startsWith("Bearer ")) {