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>

package.json

@@ -0,0 +1,7 @@
+{
+  "name": "@groombook/api",
+  "version": "0.0.1",
+  "private": true,
+  "type": "module",
+  "license": "AGPL-3.0-only"
+}