Complete Story 1-1 and fix TypeScript issues

Add TypeScript strict mode and typecheck tasks to monorepo infrastructure.
Fix E2E test @payload-config alias and frontend TypeScript errors.

- Add tsconfig.json to backend with strict mode and path aliases
- Add typecheck task to Turborepo and all packages
- Fix @payload-config alias for E2E tests and dev server
- Add setToken method to AuthService for middleware use
- Fix implicit any types in Footer.astro and Header.astro
- Remove invalid typescript config from astro.config.mjs

apps/backend/vitest.config.mts

@@ -1,9 +1,15 @@
 import { defineConfig } from 'vitest/config'
 import react from '@vitejs/plugin-react'
 import tsconfigPaths from 'vite-tsconfig-paths'
+import path from 'node:path'
 
 export default defineConfig({
   plugins: [tsconfigPaths(), react()],
+  resolve: {
+    alias: {
+      '@': path.resolve(__dirname, './src'),
+    },
+  },
   test: {
     environment: 'jsdom',
     setupFiles: ['./vitest.setup.ts'],