Improve request logging and make health check lightweight

- Log slow requests (>1s) at WARNING level with slow=true flag
- Log request timeouts at WARNING level in TimeoutMiddleware
- Replace heavy GetStatsContext with lightweight Ping in health check
- Add Ping method to database interface (SELECT 1)

internal/database/database.go

@@ -1997,3 +1997,14 @@ func (d *Database) Checkpoint(ctx context.Context) error {
 
 	return nil
 }
+
+// Ping checks if the database connection is alive with a lightweight query.
+func (d *Database) Ping(ctx context.Context) error {
+	var result int
+	err := d.db.QueryRowContext(ctx, "SELECT 1").Scan(&result)
+	if err != nil {
+		return fmt.Errorf("database ping failed: %w", err)
+	}
+
+	return nil
+}

internal/database/interface.go

@@ -88,6 +88,7 @@ type Store interface {
 	Vacuum(ctx context.Context) error
 	Analyze(ctx context.Context) error
 	Checkpoint(ctx context.Context) error
+	Ping(ctx context.Context) error
 }
 
 // Ensure Database implements Store