Initial extraction: groombook/api from groombook/app monorepo

Part of GRO-802 monorepo breakdown.

Changes:
- Extract apps/api/ as the main API service
- Inline packages/db/ (database schema, migrations, utilities)
- Inline packages/types/ (shared TypeScript types)
- Add CI workflow for lint, typecheck, test, build, docker
- Port Dockerfile with 4 stages: runner, migrate, seed, reset

Co-Authored-By: Paperclip <noreply@paperclip.ing>

packages/db/package.json

@@ -0,0 +1,38 @@
+{
+  "name": "@groombook/db",
+  "version": "0.0.1",
+  "private": true,
+  "type": "module",
+  "main": "./dist/index.js",
+  "types": "./src/index.ts",
+  "exports": {
+    ".": {
+      "default": "./dist/index.js",
+      "types": "./src/index.ts"
+    },
+    "./factories": {
+      "default": "./src/factories.ts",
+      "types": "./src/factories.ts"
+    }
+  },
+  "scripts": {
+    "build": "tsc",
+    "generate": "drizzle-kit generate",
+    "migrate": "drizzle-kit migrate",
+    "seed": "tsx src/seed.ts",
+    "reset": "tsx src/reset.ts && drizzle-kit migrate && tsx src/seed.ts",
+    "studio": "drizzle-kit studio",
+    "typecheck": "tsc --noEmit"
+  },
+  "dependencies": {
+    "drizzle-orm": "^0.38.4",
+    "postgres": "^3.4.5"
+  },
+  "devDependencies": {
+    "@types/node": "^22.10.7",
+    "drizzle-kit": "^0.30.4",
+    "tsx": "^4.19.0",
+    "typescript": "^5.7.3"
+  },
+  "license": "AGPL-3.0-only"
+}