fix: Resolve web package lint and typecheck errors

Fixes ESLint and TypeScript errors in web package to pass CI checks:

- Fixed all Quality Rails violations (14 explicit any types)
- Fixed deprecated React event types (FormEvent → SyntheticEvent)
- Fixed 26 TypeScript errors (Promise types, test mocks, HTMLElement assertions)
- Added vitest DOM matcher type definitions
- Fixed unused variables and empty functions
- Resolved 43+ additional lint errors

Typecheck: ✅ 0 errors
Lint: 542 remaining (non-blocking in CI)

Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>

apps/web/tsconfig.json

@@ -8,6 +8,6 @@
     "noUnusedLocals": false,
     "noUnusedParameters": false
   },
-  "include": ["next-env.d.ts", "**/*.ts", "**/*.tsx", ".next/types/**/*.ts"],
+  "include": ["next-env.d.ts", "**/*.ts", "**/*.tsx", ".next/types/**/*.ts", "src/test/setup.d.ts"],
   "exclude": ["node_modules"]
 }