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>

apps/api/package.json

@@ -0,0 +1,42 @@
+{
+  "name": "@groombook/api",
+  "version": "0.0.1",
+  "private": true,
+  "type": "module",
+  "scripts": {
+    "dev": "tsx watch src/index.ts",
+    "build": "tsc",
+    "start": "node dist/index.js",
+    "lint": "eslint src --ext .ts",
+    "typecheck": "tsc --noEmit",
+    "test": "vitest run"
+  },
+  "dependencies": {
+    "@aws-sdk/client-s3": "^3.800.0",
+    "@aws-sdk/s3-request-presigner": "^3.800.0",
+    "@groombook/db": "workspace:*",
+    "@groombook/types": "workspace:*",
+    "@hono/node-server": "^1.13.7",
+    "@hono/zod-validator": "^0.7.6",
+    "better-auth": "^1.5.6",
+    "hono": "^4.6.17",
+    "node-cron": "^3.0.3",
+    "nodemailer": "^6.9.16",
+    "stripe": "^22.0.0",
+    "telnyx": "^1.23.0",
+
+    "zod": "^4.3.6"
+  },
+  "devDependencies": {
+    "@types/node": "^22.10.7",
+    "@types/node-cron": "^3.0.11",
+    "@types/nodemailer": "^6.4.17",
+    "@vitest/coverage-v8": "^3.2.4",
+    "eslint": "^9.18.0",
+    "tsx": "^4.19.2",
+    "typescript": "^5.7.3",
+    "typescript-eslint": "^8.20.0",
+    "vitest": "^3.2.4"
+  },
+  "license": "AGPL-3.0-only"
+}