Add comprehensive error logging and monitoring

[clencyc]

Description

Add structured logging and error tracking to help debug issues in production.

Acceptance Criteria

• All endpoints log requests/responses
• Errors logged with full context (request, user, video info)
• Error tracking service integrated (Sentry or similar)
• Log levels: DEBUG, INFO, WARNING, ERROR, CRITICAL
• Logs in JSON format for easier parsing
• No sensitive data in logs (API keys, file paths)
• Performance: logging overhead < 5%
• Configuration in .env

Implementation Details

• Use Python logging module with JSON formatter
• Create logger utility in LiveEditBackend/utils/logger.py
• Integrate with Sentry for error tracking (optional but recommended)
• Add middleware to log all requests
• Structure: timestamp, level, endpoint, error_message, traceback

Logging Checklist

• All API endpoints
• Video processing steps
• Gemini API calls (success/failure)
• Database operations
• File operations

Difficulty: Medium

Estimated time: 3-4 hours