feat: add automatic database initialization and recovery

- Add init_database() with alembic programmatic API and retry logic
- Add connection retry with exponential backoff (5 attempts)
- Improve error messages for connection/auth failures
- Add table existence check before seeding data
- Update startup event to run migrations before seeding
- Add wait-for-db.sh script for Docker containers
- Update Docker and docker-compose configurations

Quality gates: ruff ✓, mypy ✓, unit tests (8 passed)

openspec/changes/database-setup-recovery/specs/database-initialization/spec.md

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+## ADDED Requirements
+
+### Requirement: Automatic Database Initialization
+
+The system SHALL automatically initialize the database on application startup.
+
+#### Scenario: Fresh database
+- GIVEN a new database with no tables
+- WHEN the application starts
+- THEN it runs all pending migrations
+- AND creates all required tables
+- AND seeds built-in data
+- AND starts accepting requests
+
+#### Scenario: Database with existing migrations
+- GIVEN a database with some migrations applied
+- WHEN the application starts
+- THEN it runs only pending migrations
+- AND does not re-run existing migrations
+
+### Requirement: Database Connection Resilience
+
+The system SHALL retry database connections during startup.
+
+#### Scenario: Database not ready
+- GIVEN the database is not yet accepting connections
+- WHEN the application starts
+- THEN it retries the connection 5 times
+- AND waits 2 seconds between retries
+- AND fails gracefully with a clear error message
+
+#### Scenario: Database connection refused
+- GIVEN the database is unreachable
+- WHEN the application starts
+- THEN it logs a clear error: "Database connection failed"
+- AND exits with a non-zero status code
+
+### Requirement: Seed Data Management
+
+The system SHALL handle seed data after migrations complete.
+
+#### Scenario: Seed after migrations
+- GIVEN migrations have just been applied
+- WHEN seeding built-in tool types
+- THEN the seeding only runs after migrations succeed
+- AND handles missing tables gracefully
+
+### Requirement: Startup Error Messages
+
+The system SHALL provide clear error messages for common database issues.
+
+#### Scenario: Missing migrations
+- GIVEN tables are missing because migrations haven't run
+- WHEN the application starts
+- THEN the error message indicates: "Database not initialized. Run migrations."
+
+#### Scenario: Authentication failure
+- GIVEN database credentials are wrong
+- WHEN the application starts
+- THEN the error message indicates: "Database authentication failed"