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/src/lib.rs

@@ -6,11 +6,11 @@ use axum::{
 use tower_http::cors::{CorsLayer, Any};
 use std::sync::Arc;
 
-mod auth;
-mod models;
-mod handlers;
-mod calendar;
-mod config;
+pub mod auth;
+pub mod models;
+pub mod handlers;
+pub mod calendar;
+pub mod config;
 
 use auth::AuthService;