feat: add database mode for LLM message storage and analytics

Implements a database mode that stores LLM prompts and responses in PostgreSQL
instead of making API calls. This enables:
- Testing without consuming API credits
- Collecting analytics on usage patterns
- Development and debugging workflows

Changes:
- Added DatabaseClient class for PostgreSQL operations
- Modified LLMAnalyzer to support database/API mode toggle
- Added USE_DATABASE config flag to switch between modes
- Included Docker Compose setup for PostgreSQL
- Added utility scripts for database init and analytics viewing
- Comprehensive documentation in DATABASE_MODE.md

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

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

requirements.txt

@@ -5,3 +5,4 @@ requests
 pytest
 pytest-mock
 openai
+psycopg2-binary