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>

README.md

@@ -0,0 +1,38 @@
+# GroomBook API
+
+GroomBook API service — extracted from the [groombook/app](https://github.com/groombook/app) monorepo.
+
+## Overview
+
+This repository contains the GroomBook API service, including:
+- REST API endpoints
+- Database schema and migrations (via Drizzle ORM)
+- Authentication (via Better Auth)
+- Background job handlers
+
+## Structure
+
+```
+apps/api/        # API service source
+packages/db/     # Database schema, migrations, and utilities
+packages/types/  # Shared TypeScript types
+```
+
+## Setup
+
+```bash
+pnpm install
+cp .env.example .env  # Fill in required environment variables
+pnpm --filter @groombook/api dev
+```
+
+## Docker
+
+```bash
+docker build -t ghcr.io/groombook/api:latest .
+docker run -p 3000:3000 ghcr.io/groombook/api:latest
+```
+
+## License
+
+AGPL-3.0-only