feat: Implement database connection status API

[ApostolisFalaras]

Description

Expose backend and DBMS availability information required by the frontend connection/status indicators.

Tasks

• Add GET /api/health
• Report backend availability
• Report DBMS process state
• Report active database name
• Report database connection state
• Return lightweight status information only

Acceptance Criteria

• The frontend can determine whether the backend is reachable.
• The frontend can determine whether the DBMS process is available.
• Database status can be refreshed without executing a SQL query.

Depends On

• feat: Implement DBMS process manager