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/src/components/auth/LogoutButton.tsx

@@ -16,7 +16,7 @@ export function LogoutButton({ variant = "secondary", className }: LogoutButtonP
   const handleSignOut = async () => {
     try {
       await signOut();
-    } catch (_error) {
+    } catch (error) {
       console.error("Sign out error:", error);
     } finally {
       router.push("/login");