Reorganize repo: backend/, deploy/, docs/ layout + AGENTS.md

Restructure the monorepo into clear top-level directories:
- backend/: Go module root (cmd, internal, pkg, configs, migrations,
  docs/swagger, scripts, tests, go.mod, Makefile, .air.toml)
- deploy/: Docker (Dockerfile, docker-compose*), quickstart, fluentd
- docs/: project documentation + reports/ (moved from repo root)
- AGENTS.md: new AI coding-agent guide at repo root

Update all references to the new layout:
- Dockerfile: COPY backend/go.mod, COPY backend/ (context = repo root)
- docker-compose files: context ../.., dockerfile deploy/docker/Dockerfile,
  env_file ../../.env, volume mounts ../../backend:/app
- deploy/quickstart/compose.yaml: dockerfile deploy/docker/Dockerfile
- CI: working-directory: backend for go commands, file deploy/docker/Dockerfile,
  coverage path backend/coverage.out, health_check backend/scripts/
- backend/Makefile: docker target uses -f ../deploy/docker/Dockerfile ../
- README: architecture tree, quickstart, config paths updated

Move root stray scripts (rename_models.*, run_m11_tests.sh, verify_build.sh,
gorm_bool_main.go) to backend/scripts/legacy/. All moves via git mv to
preserve history. Build, vet, SQLite tests, and docker compose config verified.

backend/internal/app/database.go

@@ -0,0 +1,61 @@
+package app
+
+import (
+	"fmt"
+	"time"
+
+	"gorm.io/driver/postgres"
+	"gorm.io/gorm"
+	gormlogger "gorm.io/gorm/logger"
+
+	"github.com/gochat/gochat/internal/config"
+	applogger "github.com/gochat/gochat/pkg/logger"
+)
+
+// NewDatabase creates a GORM database connection.
+// Pattern follows Chatwoot's ActiveRecord PostgreSQL setup in config/database.yml
+func NewDatabase(cfg *config.DatabaseConfig, logLevel string) (*gorm.DB, error) {
+	dsn := cfg.DSN()
+	
+	// Map log level to GORM logger level
+	var gormLogLevel gormlogger.LogLevel
+	switch logLevel {
+	case "silent":
+		gormLogLevel = gormlogger.Silent
+	case "error":
+		gormLogLevel = gormlogger.Error
+	case "warn":
+		gormLogLevel = gormlogger.Warn
+	case "info":
+		gormLogLevel = gormlogger.Info
+	default:
+		gormLogLevel = gormlogger.Info
+	}
+
+	db, err := gorm.Open(postgres.Open(dsn), &gorm.Config{
+		Logger:                 gormlogger.Default.LogMode(gormLogLevel),
+		PrepareStmt:            true,
+		DisableForeignKeyConstraintWhenMigrating: true, // migrate first, add FK constraints later
+	})
+	if err != nil {
+		return nil, fmt.Errorf("failed to connect to database: %w", err)
+	}
+
+	// Configure connection pool (ref: Chatwoot config/database.yml pool settings)
+	sqlDB, err := db.DB()
+	if err != nil {
+		return nil, fmt.Errorf("failed to get underlying sql.DB: %w", err)
+	}
+	sqlDB.SetMaxOpenConns(cfg.MaxOpenConns)
+	sqlDB.SetMaxIdleConns(cfg.MaxIdleConns)
+	sqlDB.SetConnMaxLifetime(5 * time.Minute)
+	sqlDB.SetConnMaxIdleTime(10 * time.Minute)
+
+	// Set default statement timeout (ref: Chatwoot POSTGRES_STATEMENT_TIMEOUT)
+	db.Set("gorm:statement_timeout", "5000ms")
+
+	applogger.L().Infof("Connected to PostgreSQL: %s@%s:%d/%s (pool=%d)",
+		cfg.User, cfg.Host, cfg.Port, cfg.DBName, cfg.MaxOpenConns)
+
+	return db, nil
+}