Add comprehensive integration tests for all API endpoints

- Add 9 integration tests covering all HTTP endpoints
- Test authentication flow: login, verify, user info
- Test calendar operations: create, delete, list events
- Test event CRUD: create, update, delete, refresh
- Add TestServer utility for automated server setup
- Test both success and error scenarios (401, 400, 404)
- Integration with real CalDAV server using .env credentials
- Fix test module visibility by making handlers public
- Move misplaced unit tests into proper test module
- All tests passing: 7 unit + 9 integration = 16 total

🤖 Generated with [Claude Code](https://claude.ai/code)

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

backend/Cargo.toml

@@ -34,4 +34,7 @@ base64 = "0.21"
 thiserror = "1.0"
 
 [dev-dependencies]
-tokio = { version = "1.0", features = ["macros", "rt"] }
+tokio = { version = "1.0", features = ["macros", "rt"] }
+reqwest = { version = "0.11", features = ["json"] }
+tower = { version = "0.4", features = ["util"] }
+hyper = "1.0"